"when President Kennedy became convinced that the Soviets had indeed installed offensive missiles in Cuba, he was privately furious"
Now, just because Kennedy perceived disrespect does not mean that was Khrushchev’s world view. There’s a case to be made that Khrushchev simply viewed it as a cheap, readily available way to reach approximate nuclear parity. That if we the U.S. had missiles in Germany and Turkey with short flight times to Moscow, then the Soviets wanted missiles with the same short flight time to Washington.
Actually it was pretty complicated. Certainly the USA was hellbent on overthrowing Castro for one thing, and certainly a major purpose in putting missiles in Cuba was to demonstrate that the Soviet tripwire protecting Cuba was no empty rhetoric--it would be clear to Americans that invading Cuba when operational Soviet missiles manned by Soviet officers were placed there would
very likely trigger a general Soviet nuclear response.
But emotion definitely came into it--on both sides. The fact is, the Jupiter missile deployment to Turkey (and Italy) did somewhat erode the Soviet security situation, but not to any existential degree--it's not like the USSR would surely survive and win without the Jupiters threatening them and would surely collapse and lose with them. The Jupiters were pretty marginal, basically second or third string and obsolete; a USAF officer testified they "could be taken out by one guy with a rifle." The purpose the Kennedy Administration had in supplying them to these allied states was not to gain a a great strategic advantage, but the political purpose of reassuring the governments of Italy and Turkey that the USA was deeply committed to their defense. In fact
exactly the same reason that was one of several the Soviets had for putting missiles in Cuba!
But when Khrushchev was informed of this American action in Turkey, he just happened to be vacationing on the Black Sea at the time. Those missiles were literally aimed to kill
him, was his emotional reaction. Khr. was quite an emotional creature-for that very reason I am sure cold reason was always also present in his decisions and he understood that the personal aspect was accidental--but still, he felt the move as both a personal threat and an insult to himself and the dignity of the USSR, and this red hot emotion had a lot to do with is resolve to pay back the Americans in exact kind.
Now it is also a fact that the missiles the Russians moved into Cuba were top of the line--not their biggest, but not as quaintly obsolescent as the Jupiters were, and that unlike the Jupiters they did greatly change the strategic threat to the USA--without the Cuban missiles, Soviet ICBMs based in the USSR were both few in number, generally long in response time (most being their first wave R-7 rockets, the same rocket that launched Sputink, Vostok, and to this day launches Soyuz spacecraft, which used liquid oxygen and therefore could not be kept ready to launch on standby but had to be fueled over half an hour or more, and were like the Jupiters kept exposed and vulnerable on launching pads and not sheltered in hardened silos--this meant that a successful American penetration of Soviet air defenses could take them out on their pads before they could be launched--like the Jupiters!) and of dubious reliability and accuracy. Thus, considering that Soviet airborne bombers were also quite unlikely to penetrate US air defenses and had limited range, the ability of the USSR to credibly threaten major devastation to the USA was low, whereas American nuclear assets, including superior ICBMs in greater numbers, were pretty likely to ruin the Soviet Union. Had the Cuban missiles all been made operational and left that way, the immediate threat to the USA would have been far higher, and in particular they would fill a gap that had been especially safe from anything but submarine launched missiles. So objectively speaking it was, discounting the political risks involved, a good strategic chess move to make--but one of limited duration, for the Soviets were arming themselves with superior ICBMs, less vulnerable, more powerful and most of all in much greater numbers, and pretty soon the Cuban based threat would be superfluous, as far as strategic balance considerations mattered.
Whereas, getting them into Cuba and keeping them there would most definitely improve the odds that Americans would be deterred from invading, and that permanently. This is why I think that in terms of rationally calculated strategic considerations, the Cuban missiles would
mainly serve to protect Cuba by tripwire deterrence, and the strategic benefit to the USSR itself would be temporary. Of course one can argue "temporary" was important enough in the context of two superpowers who each believe the other is liable to wish to start WWIII any day.
It remains historically true though that if we can document JFK reacting emotionally, so is Khrushchev documented to have initiated this plan of his for emotional reasons. Protecting the Cuban comrades was his first proclaimed priority, in internal Kremlin arguments, and buffering Soviet security another, but he was pretty open too that actually this was tit for tat, that Cuba was a response to the Jupiters in Turkey--and that since the Americans had done what they did in Turkey, the world community would hardly condemn the Soviets for doing an exactly parallel thing in Cuba.
However he did understand the Americans would not consider themselves bound by abstract rules of fair play, and therefore they would act to preempt his move if he announced it, so it had be accomplished in secret and presented to the world as a
fait accompli.
Now we come to the ATL question--if it were Nixon instead of Kennedy in the White House, how would things be different?
1) Will Khrushchev feel the same provocations? I daresay the Jupiter program was not a pure Kennedy initiative, that Nixon too would have regarded the same offer of IRBMs to such peripheral allies to reassure them.
2) Will a Castroite Cuba still exist to propose to defend? I am not so sure that, fully informed of the facts of Bay of Pigs plans, Nixon would have acted much differently than Kennedy did. If anything his greater possible engagement with the planners might have given him colder feet, and instead of a botched operation as OTL or the more "all in" and undeniable full scale overt US invasion people assume, he might instead have done the opposite--pulled in our horns and backed off trying to overthrow Cuba, or anyway sticking as Kennedy, Johnson and their successors did to deniable and limited if relentless attempts to screw them up. Kennedy's problem was that he let a half baked plan cooked up in the Eisenhower years go forward without taking a close look at it first--had he paid real attention he'd more likely have realized this had evolved to an untenable scheme falling between stools, and that the USA would have to go big or go home.
Or perhaps not. I think US leadership, bipartisan, badly underestimated how much support Castro enjoyed. They may have sincerely and with overwhelming consensus believed that the level of deniable force the planned operation involved would be sufficient to get a snowball of anti-Castro insurrection going, and that once that progressed enough the USA could then be called in by the invading "government in exile" to help openly. Thus Nixon the same as Kennedy might have committed the same blunder.
Then there is the question of, if the planned Bay of Pigs invasion did meet the same debacle as OTL, would Nixon unlike Kennedy double down on a developing mess by overtly tipping the US hand and bringing large numbers of US forces to bear on breaking Castroite resistance to allow the invaders to prevail? As in Vietnam, Americans might assume it was just a matter of bringing a large enough fraction of our forces to the task, with a breaking point favoring our side being inevitable. I think it is equally sure though that while that finite force required might be easily attained with spare US forces while maintaining full vigilance in Europe and other Cold War fronts, we would initially underestimate it and find our forces along with the Cuban counterrevolutionaries decimated until we escalated to a degree no one anticipated. Then indeed Castroite resistance might collapse and the swift occupation of the island nation follow--but after that, if we were to leave too soon, it would break out again, and in the interim for at least a time asymmetrical resistance to the American occupation would be bitter and costly. We might indeed, perhaps, break it and a sullen pro-capitalist, if not actually pro-Yankee, regime take over. I would predict it would be generations if ever before Cuba would have meaningful democracy; the immediate regime would be some handpicked dictator running a police state. So there wouldn't be any honest political barometer to gauge whether the Castroite dream is dead or not--and the people who supported Castro OTL would have no way of knowing whether they were better off or worse off. Pro-Yankee factions would assume they'd be worse off if Castro had remained, anti-American factions that they would have been far better off so regardless of whether ATL Cuba was richer or poorer, revolutionary sentiment, reinforced by patriotism, would simmer for generations to come. This would mean American masterminds would be reluctant to see the regime loosen up and ready to excuse any excesses or blatant corruption however plain--"remember Castro!" they'd say just as often as the Castroite underground.
But none of this contradicts the argument of "no Caribbean Crisis" because the Russians would lose their opportunity--unless either Nixon like Kennedy would think twice about escalating the invasion when rosy scenario predictions went sour, or after some higher level engagement with limited US force he foresaw the deep costs of following through and backed out. Conceivably there would be a truce at some front and there would subsequently be an "East Cuba" and "West Cuba" along the same lines as Korea and Vietnam. In any of those cases, Khrushchev would have as much or more reason to think he could get away with sneaking missiles into whatever part of Cuba the Castroites still hold--on one hand Yankee surveillance is higher, on the other the polarized situation makes for tight control of information on both sides--on the gripping hand, each side is full of natural spies for the other to be sure.
But if the Americans resolve to follow through even if the interim before Yankee conquest is complete is quite long, a matter of years, the writing will be on the wall and K would not dare risk planting Soviet missiles in territory that could be overrun any day and will be exposed to constant bombing and surveillance flights all the time.
Now we have to factor in calculations on both sides that might cause this Cuban crisis, the crisis caused by Nixon doubling down and insisting on a victorious invasion whatever it costs, to itself trigger Armageddon. It probably won't because Khrushchev and other insiders in the Kremlin would know US nuclear forces would destroy the USSR while all the Soviets could count on would be to kill a certain fraction of Americans-10, if they got real lucky maybe 20 percent. Europe would be an apocalyptic mess as would be Japan and South Korea--with American screening forces destroyed or deployed elsewhere Taiwan might be conquered by the PRC, and perhaps the North Vietnamese Army would go for broke and find the Saigon government helpless with US logistics cut off. But when the dust of the Exchange settles, the USA will still be in being though Washington DC is likely gone the way of the dodo along with NYC and a couple dozen other important cities. Fallout will lead to hungry years (along with a certain number of people being majorly poisoned by contaminated food they eat out of desperation, seriously shortening their life spans) and probable police state aspects of life in America, but it will be paradise compared to most of the rest of the former developed world, and well able to support a massive military establishment that simultaneously polices the home front and runs roughshod over the more desirable parts of the rest of the world.
Or, after some bluster and maybe some calculated loss in limited war, the Soviets back down and Nixon is left in relative peace to mire the USA in the conquest and occupation of Cuba. Just as the Soviets could afford to be second in the Space Race they can afford to be driven out of the running in some spot in the Third World, pointing to the massive efforts the Americans have to make and making the most they can politically over deplorable acts of American repression, painting the Yankees in the same light the world left regarded Franco--and his Nazi and Fascist patrons.
So, no Cuban
missile crisis, instead the 1960s are defined by the conventional Cuban Crisis. The Soviets will win diplomatic brownie points for withdrawing and sparing the world nuclear Ragnarok, all the while steadily building up the missile forces at home that can deliver a truly comprehensive Ragnarok within half a decade. Khrushchev might have greater conviction of Nixon's resolve than Kennedy's, but he will not surrender control of East Berlin or any of East Germany. The Vietnamese will face a USA that is heavily engaged in Cuba. To be sure Nixon in these circumstances might crack down on the counterculture and New Left, and bully liberals in both mainstream party into signing off on major police state powers, and with the nation in the hand of sophisticated spinmeisters from Madison avenue a spirit of gung ho patriotism might be cultivated that makes the Americans ruthless and aggressive in Vietnam to the point that something like the aggression in Cuba is extended to South Vietnam and eventually invasion of the North--as in Korea though the closer they get to the PRC the more worried American commanders and especially diplomats will be that China might suddenly go all in on the North's ostensible behalf. So, no actual conquest of Hanoi and taking on de-Communifying a second Third World do it yourself Leninist regime--perhaps Annam would be annexed to the South in full and turned into a giant DMZ; I imagine Cambodia's neutrality would go to the dogs with a coup by some pre-incarnation of Lon Nol and the exile or death of Prince Sihanouk. Because of censorship and spin control at home, the anti-war movement against the Vietnam intervention would not be allowed to grow, but service in Vietnam and probably Cambodia would have a hidden demoralizing effect on the US military, the Army suffering especially--and perhaps with notoriety of the war suppressed, it would spread to all deployments everywhere and go unchecked, reinforced by similar if less severe traumas in Cuba. Perhaps in Cuba there would be more cultural contact between occupiers and occupied; the regime controllers, American and Cuban, would try to steer US service member fraternization to vetted "good Cubans," but the controls might be more permeable than the spymasters believe and a certain number of US troops could be propagandized to leftist radicalism, along with people who arrive at such conclusions by sheer deduction. An American counterculture would be repressed and limited but become a hothouse of consensus radical leftism and this would propagate in particular in the military. It probably would be mainly concentrated in the Army to be sure.
With no Caribbean Crisis to discredit him, Khrushchev might last longer, meaning a serious effort at gaming out a detailed ATL would be needed to predict what happens to the Soviet Union when he eventually is out. Odds are he will undermine himself in other ways and be ousted not too long after he was OTL, and it seems a solid bet something like the Brezhnev period will follow--if one calls it Stalinist, definitely still Stalinist Lite, but less buoyant with optimism and more conventional and complacent than under the mercurial Khrushchev. He died in the early '70s so even if he learns to avoid alienating all his allies his personal writ is limited. I would imagine the Berlin Crisis would be much as OTL with the Berlin Wall being the eventual solution.