Perhaps they later realised that problem, and corrected it in their new version (hence why Obama’s says ‘Date filed by Registrar’ and not ‘Dated filed by State Registrar’).
I think accusing some random person’s genealogy site of faking a birth certificate of someone from their family tree for political reasons is a bit of a stretch.
My point is that there are many inconsistencies in the “documents” that people are using as evidence and that anything can be faked quite well today. Until several impartial parties verify the original I’ll remain skeptical.