The Supreme Court, as you well know, disagrees. They said this was the nomenclature known to the framers of the Constitution. Also, you DO understand that when Vattel says native (as in native born), it STILL means you have to be born to citizen parents. The framers would have understood this too. Under such an understanding of being a native or natural citizen at birth, you STILL HAVE TO HAVE citizen parents in order to be a natural citizen, a native or indigenous, etc., ... and as Vattel, John Jay, the Supreme Court, John Bingham, etc., would acknowledge ... free of foreign ties. Under all circumstances, Obama does not meet any of these definitions.
If the Founders were following Vattel, they should have followed his phrasing and used "native citizen" rather than confusing everyone by using a term so close to the common law phrase "natural born subject".
But they didn't, and the phrase they used makes sense as the American version of Natural Born Subject.
"II. The fundamental principle of the common law with regard to English nationality was birth within the allegiance, also called "ligealty," "obedience," "faith," or "power" of the King. The principle embraced all persons born within the King's allegiance and subject to his protection. Such allegiance and protection were mutual -- as expressed in the maxim protectio trahit subjectionem, et subjectio protectionem -- and were not restricted to natural-born subjects and naturalized subjects, or to those who had taken an oath of allegiance, but were predicable of aliens in amity so long as they were within the kingdom. Children, born in England, of such aliens were therefore natural-born subjects. "
They also cite Dicey:
"Natural-born British subject" means a British subject who has become a British subject at the moment of his birth." and "Subject to the exceptions hereinafter mentioned, any person who (whatever the nationality of his parents) is born within the British dominions is a natural-born British subject. This rule contains the leading principle of English law on the subject of British nationality."