Natural born is not exactly the same as native born. In fact some think it was a translation error..of vattels work.
It could be, I don’t really know.
That provision in the constitution which requires that the president shall be a native-born citizen (unless he were a citizen of the United States when the constitution was adopted,) is a happy means of security against foreign influence,
He has quite clearly indicated that the terms Natural born and Native born are interchangeable as far as article II, section I, clause 5 is concerned.