Actually, I think that the right answer depends upon the year that you are talking about. The laws were changed (within BHO’s lifetime).
Statute law cannot overrule the Constitution. All statute laws are supposed to 'ride on' the Constitution, and align with it. If a law does not, it's considered to be un-constitutional, and can be struck down on those grounds.
The definition of the Natural Born Citizen requirement in the US Constitution has never been definitively ruled on by the Supreme Court, though they have come close on a few occasions.
However, there is an argument that if ANY law was required to grant citizenship to the child, then the child was not a "natural-born citizen."
It seems to me three types of citizenship were recognized at the time the Constitution was enacted:
1. Natural-born citizenship -- this was citizenship that a child received at birth, not by operation of a grant of citizenship by a Sovereign (e.g., the Crown or a country), but by operation of the principle that no man is born state-less, therefore every man is born a citizen of his father's country. (Whether or not this was recognized as also flowing through the mother is a separate issue).
2. Citizenship at birth by operation of law --- this was citizenship that was granted at birth, but it's source was the operation of a law. I.e., the Sovereign made a law that all children born within the realm would be citizens (by the way, this was a bad thing, not a good thing -- it's purpose was so the Crown could tax, conscript and otherwise subject everyone to its jurisdiction). 3. Naturalized citizenship --- citizenship voluntarily applied for at some point after birth, according (again) to laws and standards set out by the Sovereign.
The only citizenship one was naturally born with was the citizenship of one's father. All other forms of citizenship are based on a grant of citizenship by the Sovereign, which could likewise be changed.
For example, the U.S. can change standards for citizenship at birth. But no country can change citizenship granted "naturally" by being born to a person with a certain citizenship.
But the constitution still has the same requirements.
The law of the land.