I assume it's Jefferson vs. Adams. And Jefferson might actually stand a better chance of winning than in 1796 in OTL--the Republicans did well in congressional races in 1792:
https://en.wikipedia.org/wiki/Party_divisions_of_United_States_Congresses and George Clinton came suprisingly close to upsetting John Adams for vice-president. (Clinton had been the Republicans' VP candidate only because VA electors could not consitutionally vote for both Washington and Jefferson.) Clinton was under a number of disadvantages: his Antifederalist record, the dubious nature of his victory in the April 1792 New Yotk governor's race
https://en.wikipedia.org/wiki/1792_New_York_gubernatorial_election and the lateness of the Republicans in endorsing him for vice-president. John P. Kaminski in
George Clinton: Yeoman Politician of the New Republic, p. 236, remarks on the logistical difficulties of Clinton's candidacy: "'Presidential electors were elected in mid-to-late fall and were required to cast their ballots on December 5. To inform all of these electors that Clinton was the designated Republican candidate was nearly impossible."
https://books.google.com/books?id=CzENUusPxfoC&pg=PA236 In view of this and the nonexistence of any but the loosest party organization on a national scale--the surprising thing is that:
"With their low-keyed, behind-the-scenes activity, the Republicans came close to upsetting Adams. Although Washington was again the unanimous choice of 132 electors--two electors from Maryland and one from Vermont
did not vote--Adams had only seventy-seven votes. Clinton, with all the votes of Virginia, New York, North Carolina, and Georgia, and a single vote from Pennsylvania, had a total of fifty. Kentucky cast four votes
for Jefferson, and South Carolina produced one vote for Burr.
Had Pennsylvania swung to Clinton, he would have defeated Adams." Richard P. McCormick,
The Presidential Game: The Origins of American Presidential Politics , p. 49. [emphasis added--DT]..."
Of course, the vice-presidential candidates also matter, since under the Constitution as it then existed, the man who was intended to be vice-president might become president instead. I assume that in this ATL Clinton will be the Republican vice-presidential candidate and some Southerner (a Pinckney?) the Federalist one. One surprising possibility, if the Federalists win, is the US getting its first Catholic vice-president--or even president--long before one would think it likely:
"In reply to a proposition from McHenry to support Charles Carroll as a Candidate in the event of the President's [Washington's] retirement, Hamilton observed, 'Your project with regard to the Presidency in a certain event will, I believe, not have an opportunity of being executed.-- Happily for the public tranquillity, the present incumbent, after a serious struggle, inclines, if I mistake not, to submit to another election.--If it turns out otherwise, I say unequivocally, I will co-operate in running the gentleman you mention as one of the two who are to fill the two great offices--which of the two may turn up,
first or
second, must be an affair of some casualty as the Constitution stands. My real respect and esteem for the character brought into view will ensure him my best wishes in every event.'"
https://books.google.com/books?id=3SQ-AAAAYAAJ&pg=PA59
I once asked in a soc.history.what-if post: "If Washington did decide to retire after one term, could we get John Adams/Charles Carroll vs. Thomas Jefferson/George Clinton--with the Federalist ticket winning, and with a handful of southern Federalist electors voting for the Catholic but southern (and slaveholding) Carroll
http://en.wikipedia.org/wiki/Charles_Carroll_of_Carrollton but not for the northerner Adams, so that Carroll comes out a vote or two ahead?..." Alternately, of course, anti-Catholicism could lead to a few Federalist electors not voting for Carroll, so we could get Adams-Jefferson foyr years earlier than in OTL. But it's more likely that the Federalists will name a southerner other than Carroll to be Adams' running mate.
So does the no-third-term tradition of OTL become a no-second-term tradition in this ATL? I doubt it. People would be more likely to say that General Washington was exhausted by the duties of the presidency than that he had intended to start a precedent. Indeed, even in OTL one could argue that Jefferson was the real founder of the no-third-term tradition--he was the first explicitly to make not running for a third term a principle, not just a personal choice:
"If some termination to the services of the chief magistrate be not fixed by the Constitution, or supplied by practice, his office, nominally for years, will in fact become for life; and history shows how easily that degenerates into an inheritance."
https://books.google.com/books?id=wf8PAAAAYAAJ&pg=PA458&lpg=PA458
Note though that in 1805 he did leave a small loophole, though quickly dismissing it as "impossible":
"...My opinion originally was that the President of the U.S. should have been elected for 7. years, & forever ineligible afterwards. I have since become sensible that 7. years is too long to be irremovable, and that there should be a peaceable way of withdrawing a man in midway who is doing wrong. The service for 8. years with a power to remove at the end of the first four, comes nearly to my principle as corrected by experience. And it is in adherence to that that I determined to withdraw at the end of my second term. The danger is that the indulgence & attachments of the people will keep a man in the chair after he becomes a dotard, that reelection through life shall become habitual, & election for life follow that. Genl. Washington set the example of voluntary retirement after 8. years. I shall follow it, and a few more precedents will oppose the obstacle of habit to anyone after a while who shall endeavor to extend his term. Perhaps it may beget a disposition to establish it by an amendment of the constitution. I believe I am doing right, therefore, in pursuing my principle. I had determined to declare my intention, but I have consented to be silent on the opinion of friends, who think it best not to put a continuance out of my power in defiance of all circumstances. There is, however, but one circumstance which could engage my acquiescence in another election, to wit, such a division about a successor as might bring in a Monarchist. But this circumstance is impossible. While, therefore, I shall make no formal declarations to the public of my purpose, I have freely let it be understood in private conversation..."
http://www.let.rug.nl/usa/presidents/thomas-jefferson/letters-of-thomas-jefferson/jefl167.php