But We do recognize another nation's claim on their citizens.
Which has no bearing on what I wrote. If another nation claims a person in their citizenry, that is irrelevant with regards to our claim. Does a claim by Canada on your personage as a citizen now make you not natural-born?
In fact, why do so many in this very thread argue that the parentage and claim of another nation on the offspring matters for natural born status, yet accept that James K. Polk was the first "natural born citizen" President? Both his parents were British subjects while he was born within the United States after the birth of the nation, and thus by the arguments here Polk would not be a natural born citizen.
So if we do say another nation's claim on a person matters (regardless of what our nation - or the individual - considers about that claim) then clearly many of our earliest Presidents were NOT natural born, as they came from the perpetual citizenship of Britain.
On the other hand, if we ignore (rightly) what another nation claims about one of our citizens, it's a non-issue altogether.