Those words irrefutably establish the fact that the Framers were worried about the undivided loyalty of the President, and thought that the requirement that he be a natural born citizen would be sufficient to prevent anyone with foreign allegiance (anyone who could be claimed as a subject or citizen of a foreign sovereign) from serving as President. But how could that be, if "natural born citizen" differs from "natural born subject" solely in the difference between a subject and a citizen? A British "natural born subject" could have multiple nationalities, and owe allegiance to multiple sovereigns. And many nations claim anyone with at least one parent (sometimes it must be the father, sometimes it must be the mother, sometimes both) who is a citizen or subject of that nation as a citizen/subject also.
Therefore, it is beyond any possibility of dispute that the only way the "natural born citizen" requirement can prevent a person from having allegiance to a foreign sovereign is if its meaning is the same as the one de Vattel defined and labelled "les naturel, ou indigenes," and which a professional translator translated into English as "natural born citizen" just a few short years after the "natural born citizen" requirement was written and ratified in the new US Constitution. Literally and normatively, the words "les naturel, ou indigenes" mean "the natural ones, the natives." So why did the translator render them into English as "natural born citizen," unless it was his expert opinion that the meaning of "natural born citizen" in the Constitution matched the meaning of the concept defined by de Vattel, where de Vattel specifies the purest form of citizenship as requiring both jus soli and jus sanguinis—with BOTH parents being citizens?
Clearly, if both your parents are citizens (or subjects) of the same sovereign, and you were born in that same sovereign's territory, then and only then is it impossible for any foreign sovereign to have a claim to your allegiance under the law of nations as commonly understood. John Jay's request to Washington makes no sense otherwise. If that reasoning is sound, then "natural born citizen" must have been intended to have the same meaning as de Vattel defined for his term-of-art phrases "les naturels, ou les indigenes."
No, those words establish that the Framers (in particular, John Jay) were concerned about the “admission of foreigners into the administration of our national government.” By my reading of Jay’s letter, they were concerned about the possibility of a foreigner immigrating to the U.S., becoming a naturalized citizen, and then running for President. Madison (the father of the Constitution) said it himself - in the U.S., place, not parentage, matters.