In both cases, the son born could be American or British. A natural born citizen can be no other.
A son born here of American parents can be nothing but American. No other country on the planet can call him a citizen at the time of his birth.
This is what makes him a natural born citizen as opposed to a citizen.
It is also why the “Citizen at the time of the adoption of the constitution” is in the text, not one of the founders was a natural born citizen.
Any country, including the US, can grant citizenship to whomever they want.
We could pas a law saying the next child Putin fathers is a US citizen at birth, and the child would be.