“So what?”
Natural born citizen is not found in Vattel, only in a poor translation. The French refers to the natives, or indigenous people, not “natural born citizen”.
So someone reading Vattel in the French, or using one of the English translation available at the time the Constitution was written would NOT use Vattel as a source for NBC.
“The child of the citizen is “natural born” while the child of the alien is “just a much a citizen”.”
Actually, under normal English, “The child of an alien, if born in the country, is as much a citizen as the natural born child of a citizen, and by operation of the same principle” says that as regards citizenship, there is no difference between the child of an alien born in the US, and the “natural born child” of citizens. Thus the citizenship of a natural born child of citizens and the child of an alien is of equal merit.
It doesn’t distinguish between them, for one “is as much a citizen...by operation of the same principle”.
An apple is as much a piece of fruit as an orange, but an apple is not an orange. By that principle, Obama is a fruit.
Then why the difference in terms?
They are quite clearly different or they would state it differently to mean the same.
The child of a citizen may very well be “natural born” with different, inalienable and unalterable rights that are quite distinguishable from a “just as much citizen” and that citizen may also have all the other rights of a Natural Born Citizen but not the one distinguishing characteristic of their birth or Constitutional requirement, that they be Natural Born Citizens.
In other words, a simple citizen cannot be President but a Natural Born Citizen can.
Other than that, the law treats them the same.
“The child of the citizen is “natural born” while the child of the alien is “just a much a citizen”.”
He used two words, "Naturels" and "Indigenes". The Royal Dictionary from near the period in question, indicates that the two words may *both* be translated as "naturals", but Naturels may also be translated as "natives". The way they are used in both the original French and the way the translated terms are used, it's clear they are being used as synonyms. Thus natives and naturals, but used to refer to the "citoyn" or "citizen" in the previous sentence. Thus "natives or natural born citizens is not a bad translation. It's certainy better than the one which left "indigenes" untranslated. When it was finally translated in the 1793 eddition, it was not translates as "indignious" but, as one would expect from the dictionary, as "natural" born. Our founders did not need the earlier translation. They could read the original French for themselves. Dr. Franklin was particularly adept in French, having found it useful with the French ladies, some of whom were quite well educated, when he was representing the "rebels" there.