> True, though there is no clear consensus on what the former means, it is clear it is narrower than the latter.

No the consensus is pretty clear and only gets muddied up when there are political reasons for trying to invalidate someone's presidential candidacy. To become a citizen involves going through a "naturalization" process. A natural-born citizen is anyone whose citizenship came by birth rather than through naturalization. There may have been some ambiguity when the Republic was young, borders were porous, and documentation to track births and deaths and such was scarce or easily forged. But today we've got that all pretty much locked down.