I used to research this issue quite a lot, and I believe I have figured out exactly what happened, and why so many people became confused over the years about the meaning of "natural born citizen."
But before I get into any of that, the first thing I would ask people to do is to explain why the Colonies gave up the word "Subject", and thereafter used the word "Citizen."
Why did they do this? "Subject" was the normal term for all of their history. From whence did this "Citizen" word emerge?
Interesting question. At least the term is not "Comrade." Yet.
In your example, Thomas Jefferson probably by habit initially used the term "subject" in his draft of the Declaration of Independence since the colonists had indeed been subjects of the king. Jefferson also used the term "natural subject" in his 1783 words describing English law that I posted to start this thread. But now the colonists were going to be free of owing allegiance to a king, and they would possess rights and privileges and the ability to vote for the people in charge of their government(s). So, a term other than "subject" was appropriate.
You can find similar descriptions of "subject" and "citizen" in the 1828 Webster's Dictionary [Link].
I imagine "citizen" originally meant inhabitant of a city. From the internet [Link]:
Origin of citizen
1275–1325; Middle English citisein < Anglo-French citesein, Old French citeain, equivalent to cite city + -ain -an; Anglo-French s perhaps by association with deinzain denizen
The French also switched to the term "citizen" for describing themselves during the French Revolution when they overthrew their king.