You base your whole argument on a book written by a French author in 1758. I do not dispute that the founders were familiar with it. They were familiar with many different authors, but that does not give these authors Constitutional authority. The founders created two classes of citizenship, natural born and naturalized. There is no other. If a person is a citizen and was never naturalized, then they are a natural citizen.
Then.........why in the world does the Constitution make the distinction? Here it is!
[Article II; Section I; No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.
There is no such requirement for Judges, Representatives or Senators. Why the distinction?
Because..... like it was commonly understood in the eighteenth century....a "Natural Born Citizen" was one whose parents were both citizens as well at the time of his birth. Otherwise.....don't include the phrase.... "Natural Born". Just say....."citizen".
The reason the framers insisted the phrase be included was because John Jay.....the first Supreme Court Chief Justice, had asked George Washington to make sure it was included. He felt the Chief Executive should be a person with no possible other allegiances. As with Barack Obama....it can be said he has a definite allegiance to his Father's country, Kenya whether or not he was born there. His father was not a U.S. citizen so it would be natural to also have an affinity for his father's homeland. This was what John Jay was attempting to avoid. He wanted to make sure the Commander in Chief had no such foreign entanglements. John Jay had also written many of the Federalist Papers and carried much influence among the framers of the Constitution.
It's actually not my argument. It's history.... but many folks have not been taught the Constitution during the last fifty years or so. When I was in grade school during the fifties we were taught what a "Natural Born Citizen" meant. The fact that government schools have not properly educated children in the Constitution does not make it my argument.
Again......if there is no difference in the two qualifying factors.....why spell them out? Use your head. The framers wanted a chief executive born of parents who were also citizens. That's why they specify, "At the time of the adoption of this Constitution"! Later on.....there would be no reason to worry about this qualifying factor as multitudes of qualified candidates would have been born to citizen parents. In 1789.....that wasn't the case.