The key word is "apparently". And most birtherism doesn't rely on that law, which doesn't apply in any case.

I'm going to dig around some but while the poster is waiting he might want to read a post I stuck up when this shit was just getting started, which is to say in Dec 2008, right after the election.

Nationality Act of 1940: "Wingnuttia is ablaze"

There are three basic strains of Birtherism with variations and cross-infections:
1) Obama was born in Kenya
2) Obama's mother's marriage to Soetoro and relocation to Indonesia abrogated Obama's citizenshop
3) Although is by birth a citizen, he is not a Natural Born Citizen (lots of variations here)

All of them are fatally flawed they generally have some plausible line of legal precedents that seemingly back them up. But the clear problem in each case is that people are desperately trying to start from a conclusion: that Obama was not lawfully ever a President and that it is the responsibility of the Birther's to find a reason why. Which is why they slide from one reason to another when cornered.

For example we had a sequence: Obama refuses to release Birth Certificate. When said certificate was shown to be online, the claim shifted to say it was obviously forged because of A, B or C. When Open Secrets showed that it was not forged and clearly met the test of A, B and C the claim shifted to 'It is not a Birth Certificate, it is a Certificate of Live Birth' (you will hear the term 'Long form' in connection with this). Ultimately that line of argumentation largely died out leaving people like Orly pushing the actual Kenyan birth backed up by this argument about his mother being less than five years in residence in the U.S. prior to the date of birth after turning 14.

All the arguments are bogus but the support ranges from bat-shit delusional (Orly Taitz) to standard conspiracy theory level where the supporting documentation is piled high and people can talk in mind-numbing detail about trajectories and rates of fire (Grassy Knoll).

In Dec 2008 there were reports that the Supreme Court had taken up the case of D'Onofrio v Wells. It hadn't but it gave hope to Birthers (and sparked my post linked above). After that hope was dashed remaining Birthers lashed themselves to an ever changing series of arguments, and at each stage insisting that they "just wanted to see" something or other, only after being shown it going on to "just want to see" something else. Any claims that these people are just looking for the truth pretty much belied by their compulsive clinging to a conclusion no matter what you show them.
3)