BTW, this is not a peculiarly Constitutional question. This is a question of how we will define natural born citizen, and I think that question has been answered by two events. The accession of Barack Obama to the Presidency despite his obvious lack of eligibility by the standard heretofore accepted, and two, 8 US Code, both 1401 - and 1101 (23) "The term 'naturalization' means the conferring of nationality of a state upon a person after birth, by any means whatsoever."
Not 'at birth'.
No, it doesn't. If being born a citizen pursuant to one of the subsections of 8 U.S.C. s. 1401 relating to births outside the U.S. made a person a natural born citizen in the Constitutional sense, then Mario Bellei would have won his case. He lost because he did not fall within the "Constitutional definition" of citizen per the 14th Amendment.
BTW, this is not a peculiarly Constitutional question.
I know you dearly want to bring in the U.S. Code to answer the Constitutional question. I'm merely pointing out the flaw in your argument. How Congress characterizes a matter isn't binding on the Supreme Court. Congress may, for example, impose a certain statutory penalty and indicate "it's not a tax" while the SCOTUS may still say "yes, it is."
In this case, Supreme Court over 100 years ago stated that persons born abroad who are citizens at birth are naturalized citizens, and the courts (both the SCOTUS and lower courts) have consistently affirmed that principle as recently as 1998. A statute enacted along the way defining naturalization as a process occurring after birth hasn't and doesn't trump that.
The accession of Barack Obama to the Presidency despite his obvious lack of eligibility by the standard heretofore accepted
What standard was that? I can point you to law review articles from 1968 and 1988 that state that the jus soli principle the U.S. adapted from England is settled law, and thus all native born citizens are natural born citizens. The courts and scholarly community over the past 75 years or so (at the least) have been unanimous on this point. Obama's case falls within that accepted principle.