Notice in your first quote, Alexander Hamilton suggests “be born a Citizen of the United States”. That suggests their concern was someone be born in the US, and not foreign born.
John Jay is concerned with “a strong check to the admission of Foreigners into the administration of our national Government” - and thus a natural born citizen is [or at a minimum, could refer to] one who is not foreign born.
I’ve tried discussing the list of cases you cite, but couldn’t find their text. I found The Venus, but couldn’t find ‘natural born citizen’ in the text anywhere. Perhaps my computer missed it...
I also found SHANKS V. DUPONT, 28 U.S. 242, 245, but it includes it only 2 times, once stating that the lady in question was a NBC, and this:
“There is indeed one prominent difficulty hanging over this argument which it is impossible to remove. If it proves any thing, it proves too much, since the inference, if resulting at all, must extend to put off one’s allegiance, as well to adopted citizens as to natural born citizens, and to all times and all circumstances.”
Thus, it once again contrasts natural born with adopted (naturalized) citizens.
The McCain resolution, of course, was referring to someone born outside of the USA, and I agree - if Obama was born outside the USA, he is not a NBC.
Natural born citizenship goes further. Born in country, to citizen parents.