As far as I know there are only two types of US citizen these days (assuming there are no survivors from the time of the adoption of the Constitution). One is “naturalized”, where you were not born a citizen but became one later. That kind of citizen can not become president. The other kind of citizen (including Barry) IS eligible to become a president. There isn’t a third flavor of citizen.
They are currently afforded citizenship status, but are not naturalized.
By the way, from where do you derive your definition of "natural born Citizen?"