But to be fair, what you meant to say is that there are strong analogies between the meanings of subject and citizen, and between the meanings of "natural born subject" and "natural born citizen." No one disagrees with that. The point of contention is where the similarities stop and the differences begin. Analogies are never perfect, for otherwise they would cease to be analogies at all.
de Vattel defined what he meant by "les naturels" and "indigenes" in the text itself. He was plainly and beyond possibility of dispute explaining and defining a concept. That concept was defined ostensively as the set of all persons both born in a country and born to parents both of whom were citizens. The words he wrote make it plain that he was assigning the terms "les naturels" and "indigenes" as two equivalent referents (labels) to his concept. In doing so, he made it explicit that the normative meanings of "indigene" (native) and "naturel" (natural) were not what he intended. He was defining "les naturels" and "indigenes" as terms of art in the domain of international law, and not using them normatively.
You're quite right that de Vattel did NOT use the phrase "sujets naturels." Why should he, when that phrase's normative meaning was NOT the one he intended to use, convey or define? It's precisely because the concept he was defining was NOT the normative meaning of "sujets naturel" that he didn't use that term!
You're also quite right that he explicitly stated that the concept of purest citizenship he was defining (the name doesn't matter yet) was not used in England. So if the Constitutional Convention was relying either on the law of nations in general, or on de Vattel's description of it, when they wrote "natural born citizen," then that would be yet another proof that "natural born citizen" and "natural born subject" differ in their meanings in ways other than the differences between a subject and a citizen.
Which brings us right back to where we started: Did the Constitutional Convention intend to leverage the meaning of "natural born subject" by the term "natural born citizen," with the only difference being the one between a subject and a citizen? Or not?
New words and terms of art are invented all the time. And the meaning of words and phrases change over time. Such changes do not occur due to inscrutable, random processes, nor because of error in teaching children the native language of their culture. The meanings change because the old meanings lose their utility, and new meanings gain utility, because the society and culture themselves change. Sometimes the word or phrase will be slightly altered to help disambiguate the old and new meanings, but not always.
The context and situation of the United States and Britain were not the same. Britain was ruled by a royal family. Only members of the royal family could assume the role of head of state and commander of the armed forces. And Britain was a world empire. None of that was true of the United States at the time. That establishes a compelling motive for a term that would be more restrictive than "natural born subject" in order to limit who would be eligible to become President of the United States, since the British definition of "natural born subject" would allow persons to be President who had allegiance to foreign powers. The point being that, with such a compelling motive, it would be invalid to assume without evidence that the only difference between "natural born citizen" and "natural born subject" is the one between citizen and subject.
Again, consider John Jay’s letter to Washington: “Permit me to hint whether it would not be wise and seasonable to provide a strong check to the admission of foreigners into the administration of our national government ; and to declare expressly that the command in chief of the American army shall not be given to, nor devolve on any but a natural born citizen.” That establishes the fact that the Framers were worried about the undivided loyalty of the President, and thought that the requirement that he be a natural born citizen would be sufficient to address that issue.
But how could that be, if "natural born citizen" differs from "natural born subject" solely in the difference between a subject and a citizen? A British "natural born subject" could have multiple nationalities, and owe allegiance to multiple sovereigns. The only way the "natural born subject" requirement can prevent a person from having allegiance to a foreign sovereign is if its meaning is the same as the one de Vattel defined and labelled "les naturel, ou indigenes." If both your parents are citizens (or subjects) of the same sovereign, and you were born in that same sovereign's territory, then and only then is it impossible for any foreign sovereign to have a claim to your allegiance under the law of nations as commonly understood. John Jay's request to Washington makes no sense otherwise.
If that reasoning is sound, then "natural born citizen" must have been intended to have the same meaning as de Vattel defined for his term-of-art phrases "les naturels, ou les indigenes." In which case the 1797 translation of de Vattel is no error at all, but a perfectly correct usage of terms in American English that had the exact same meaning at that time and place as intended by de Vattel.
The SCOTUS decision you cite can only set precedent with respect to issues legally before the court, which had to do with State citizenship in North Carolina, not with national citizenship of the United States. Any assertions in court decisions that don't decide the issue legally before the court are dicta, and have no value as precedent.
The operative meaning of "natural born citizen" in general, or how it applies for a putative President of the US, has never been before the Supreme Court. So there are no precedents.
"The term "citizen," as understood in our law, is precisely analogous to the term "subject" in the common law, and the change of phrase has entirely resulted from the change of government. The sovereignty has been transferred from one man to the collective body of the people, and he who before as a "subject of the king" is now "a citizen of the State."
Vattel never wrote about natural born citizens or subjects. Ever. And the passage birthers misquote was not written about English law, with the US being a subset of it, since the USA didn't exist at the time Vattel wrote.
In the USA, citizenship is primarily from birth, not parentage. Vattel, discussing other countries, had it different - but Vattel is not US law.