I think you'll find that passports only indicate citizenship, although because they include place of birth, they can be used to infer "native born" or "naturalized"... but not reliably, since persons born of parents in the service of the country (such as military or diplomatic) are considered native (and if both parents are citizens, natural) born, yet place of birth might be Nairobi, Kenya, or more likely someplace in England, Germany or South Korea.
Your second and third "classes" are subsets of the first. Both "natural born" and "naturalized" are citizens, as are native born. Natural born is a subset of "native born", while naturalized is a subset that does not intersect the set "native born". Where "native born" has the modern meaning of "born in the United States".
The reason it's not kept track of, is that all citizens have the same rights, so there is no purpose in keeping track. But being elected to office is not a right, and their are eligibility criteria which may restrict naturalized citizens from running for or holding federal elected office. For Representative and Senator, there are residency requirements, and for President a complete ban. Similarly non Natural Born citizens cannot hold the office of President, even if they are native born, and otherwise meet the eligibility requirements.
Correct me if I'm wrong: Has the United States government ever demanded that a United States citizen prove he was a natural born citizen?