So let's review. Vattel is out for two reasons:
1) He couldn't have coined the phrase "Natural Born Citizen", because he wrote his stuff in French, and the founding fathers would have needed a time machine to read the first english translation in 1797 that used that phrase.
2) The 1797 translation of Vattel (that stole the phrase "Natural Born Citizen" FROM the US constitution) was heard by SCOTUS in the case of Wong Kim Ark, and SCOTUS rejected it.
Double burn, Vattel is out.