“it means born of citizen parents, as per their own definition”
Except that definition first occurred when Vattel’s book on international law was incorrectly translated AFTER the US Constitution was passed!
The phrase in Vattel was “indigene” and “natural” - so the INDIGENOUS people are those born of two citizen parents!
I don’t know anything about Vattel’s book, but the Immigration Act of 1790 directly defines natural born as being born of citizen parents.