No one cares, or should care, what the modern usage is. And Vattel is nice, but he wrote in French. Word for word translations of terms of art might be interesting, but that's all.
We're talking late 18th century English here and the definitive reference for that and the entire history of English word usage is the Oxford English Dictionary. The entry for natural-born there makes it pretty clear that a natural-born subject did not have to be born in England, but only that his parents had to be English subjects.
ML/NJ
Don't know about you, but I'm a citizen, not a subject. So were the founders. They rejected English common law, except for some definitions. But probably not this one.
The difference is, a subject can never become the sovereign. A citizen is the sovereign. The point of the "Natural born citizen" clause is to protect the nation, that is the people. The point of the Natural Born Subject laws and court decisions was to provide more subjects for the King.