Statutory law can do just about anything. Not saying there is a third kind of citizenship, but a statute can assert any person is a citizen at birth.
See Rogers. v. Bellei. If the sole source of citizenship is statutory, even if citizenship attaches at birth, that person is naturalized without any ceremony.
If I read it correctly in that case the same statute that granted him citizenship also required he live five years in the USA before age 28, which he didn’t do. However in the case of someone born here their claim to citizenship is constitutional (the 14th amendment) not statutory. I know there is an argument about what “and subject to the jurisdiction thereof” means, but the most plain meaning would be the exclusion of children of foreign diplomats, who are not subject to our jurisdiction. Ordinary foreign citizens here are.
I think you are ignoring the fundamental difference between "born" and "naturalized". Take, for example, the wording of the citizenship clause of the 14th amendment:
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.
I would take this to mean that birth and naturalization are mutually exclusive categories.