He told us he was legally adopted. Any legal BC issued by Hawaii during his usurpation should have said "Soetoro", not "Obama".
He has yet to prove who he really is based on undisputed legal documents. Depending on the timeframe, he could be legally one or the other you mentioned above.
While he was POTUS, if his legal name was "Soetoro" and he used "Obama", then he has lied and committed fraud. If he is neither, the fraud was intentional. I'll leave it to the lawyers to decide if every document he signed while POTUS is still legally binding if the signature is fraudulent.
Whoever he is, his existence and legal name is questionable.