No, your three classes does not explain the difference between a “citizen” who is eligible for the office of Senator and Representative, and a “natural born” citizen eligible for the office of President, as the Constitution describes.
the two terms are not synonymous, and you have equated the two as equal.
A citizen is any of the three rypes mentioned in the Constitution. And we have had natural born citizens, naturalized citizens, and citizens at the time of the adoption as Senators. It is ridiculous to say that a Senator must be a fourth category of citizen- just a citizen.