The Chief Justice of the Supreme Court in the 1975 Minor versus Happersett decision cited Vattel's common law definition as part of the majority opinion, stating accordingly:
"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."
This paragraph from Justice Waite's majority opinion has never been disputed.
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.