2) The Indian Wars, Undeclared Franco-American War, and Barbary Wars stil happenbut their effects aren't as great on our history as the War of 1812 Mexican-American War and the Civil War for example Andrew Jackson isn't a national hero, neither is Zachary Taylor or Jefferson Davis or Ulysses Grant. We also don't have the positive and negative effects of the conflicts.
Pretty bad examples, actually, because two of the three are factually wrong, and the third was a hairsbreadth from passing with 2/3 majority OTL as well.
The leadup to 1812 already had a proxy war with Britain in the North West, along with sustained Indian campaigns. While the OTL votes weren't
quite your arbitrary 2/3 majority of Congress, they were extremely close. Close enough for reasonable in-time-line deviations to see it pass.
The Mexican-American war was incited by the President by putting US troops into disputed territory, and the resulting war was fought mostly with Southern militia provided by the South. Congress also approved the Declaration of War with more than enough votes, even by your higher standards. (40-2 in Senate, 173-14 in House.) Mex-Am War is a great demonstration for how even an early President could incite a war.
The Civil War, of course, is infamous for never having a declaration of war in the first place; Lincoln ordered troops in, bought stuff, and left it to Congress to retroactively authorize his actions.
So what are your examples? One case where natural politiking to gain that necessary vote is very conceivable, a case in which Congress enthusiastically went to declare a war that the President helped incite, and one in which the President acted completely without prior Congressional approval.
What we have here, already in OTL, is a tendancy for the President to lead into war either (a) with widespread Congressional support, or (B) not giving a fig leaf about a Declaration of War. It's also rather telling if you count the number of wars and conflicts the US got into, even pre-1900, without a Congressional Declaration of War. In a TL where there is a 2/3 majority requirement for formal Declarations of War,
you're going to see fewer declarations of wars, not fewer conflicts.