Bottom line: it's not definitive yet so many here claim that Vattel - and Vattel alone - is the only acceptable definition. That is simply not true, as your own reference shows.
"The Constitution does not in words say who shall be natural-born citizens. Resort must be had elsewhere to ascertain that. At common law, with the nomenclature of which the framers of the Constitution were familiar, it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also. These were natives or natural-born citizens, as distinguished from aliens or foreigners.Some authorities go further and include as citizens children born within the jurisdiction without reference to the citizenship of their parents."
And that is because children born in this country at the time from parents who were not yet citizens but in the process of naturalization were considered to be aliens and foreigners for the five years until their father took his oath of citizenship or they were old enough to take theirs.
nobody born IN the usa has to obtain any kind of citizenship application.