THe fascinating thing about Emmerich de Vattel is the false notion that he was even discussing “natural born” citizenship at all.
De Vattel was describing “Les naturels.” Some have chosen to translate this as “Natural Born Citizen.” It means nothing of the sort, but rather “Indigenous peoples.”
On occasion, “Les indigens” might be used in French, but this can have a connotation of non-Westerners in nations colonized by Westerners; I’d translate “les indigens” as “natives”; one would not call one’s own countrymen “natives” or “les indigens.”
In de Vattel’s world, if you were French, but both of your parents were Bohemians, you were Bohemian, not French. The founding fathers were defying that notion, not confirming it; if you were born in America of parents who were under U.S. law (as opposed to being aliens), you are an American. The intent of this law was to prevent citizens of European colonizing empires from moving to America and becoming president.
So the reference isn’t to “les naturels,” but to the Thomistic doctrines of natural law, which state that all men have, by nature, a nation of their own. Their nation is theirs by virtue of being natural-born, or being naturalized. Since there is no precedent in American statute, or any expression of natural law, of anyone being naturalized in a country despite being born under that country’s law (without having taken actions to renounce that citizenship, or to assume another nation’s citizenship), there is no case for asserting that someone may be a citizen by being born in a country, without being a natural-born citizen of that country.
(Statutes may exist to clarify citizenship or establishing that a territory is under U.S. law for the purposes of determining citizenship, but no-one has ever been ‘naturalized’ through such statutes.)
Vattel was referring to natural citizenship at birth. It stands to reason why this would be translated as natural-born citizens. Vattel made a distinction between citizens and subjects. Here, he is clearly talking about citizenship, which is what applies in the United States.
IMO natural-born citizen is just an alternate way of saying native-born citizen or citizen at birth.
As the article points out nobody was much concerned about various previous presidential candidates, including Romney’s father, who did not meet the strict standards proposed by some birthers.