Looks like you are trying to get “cute”, I don’t buy it! Only applies to the qualifications to hold office as President and V. President.
If the US government has in fact legally established three types of citizenship, only one of which enables a person to serve as president/vice president, you'd think that (excluding naturalized citizens) we'd be classified as natural born citizens or just plain citizens at or near the time of our birth and that our government-determined citizenship classification would appear at least on our birth certificates, our passports, our census forms, and maybe our death certificates. That's the kind of stuff the government normally likes to keep track of.
While it definitely wants us to be legally designated male or female, the US government has never bothered to legally establish which of us is natural born. That's very curious. I think it can be argued the US government's curious indifference means that it does not consider natural born a legally distinct category of citizenship.