Unfortunately, Vattel’s “definition” is NOT written in the Constitution. This is why the 14th Amendment was added. And, again, unfortunately, the 14th Amendment also did NOT completely clarify the issue with respect to the eligibility for POTUS.
The fourteenth amendments section dealing with births applied only to the children of former slaves and no one else. This there for makes that section now what is called “dead letter law” as there are no more former slaves of this nation capable of bearing children nor of being alive for that matter.