Speaking about British statutes in this subject, which are all naturalization laws.
The conflation of 'natural born' under natural law meaning the same as British natural born subjects. This is about as to the point as I have seen from a source of authority, Treatise on Citizenship. Here is conciseness that puts the kibosh on this myth:
As we see here, all of them in and out of the British dominion are put in the same basket as they are all proclaimed to be natural born subjects.
What's more "natural" than all persons are all naturally under kings? The way royalty thinks about everyone - your highness.