The term "natural born subject" was a legal term of art that originally came from natural law. Over time, it was expanded to include children born abroad to both the king and his subjects.
We used the same term in the Colonies. When we changed "subject" to "citizen," "natural born subject" changed to "natural born citizen."
The two terms were used absolutely synonymously in early America, and wherever "natural born subject" was used, it was replaced with "natural born citizen."
I hope that helps.
It comes from the theory that anyone born on the Kings land, is "naturally" owing eternal allegiance to the King. Again, let Sir Michael Foster (1762) explain it to you.
Sect. 1. With regard to Natural-born Subjects there can be no Doubt. They owe Allegiance to the Crown at all Times and in all Places. This is what We call Natural Allegiance, in Contradistinction to that which is Local. The Duty of Allegiance, whether Natural or Local, is founded in the Relation the Person standeth in to the Crown, and in the Privileges He deriveth from that Relation. Local Allegiance is founded in the Protection a Foreigner enjoyeth for his Person, his Family or Effects during his Residence here; and it Ceaseth whenever He withdraweth with his Family and Effects. Natural Allegiance is founded in the Relation every Man standeth in to the Crown considered as the Head of that Society whereof He is born a Member; and on the peculiar Privileges He deriveth from that Relation, which are with great Propriety called his Birthright. This Birthright nothing but his own Demerit can deprive Him of; it is Indefeasible and perpetual. And consequently the Duty of Allegiance which ariseth out of it, and is inseperably Connected with it, is in Consideration of Law likewise Unalienable and Perpetual.
And this is the theory of "natural law" upon which Jeff Wishes to believe US Citizenship is based. Did I mention how dumb his theory is?