I am disputing your contention that there's any meaning whatsoever in the fact that you have two different terms that apply to the same phenomenon. We found out in earlier discussions that some states don't even use the term "birth" on their "certificates" or "statements" or "certifications" or "documents" or ------------ whatever. And any given state may well use different stuff even in the same month, or simultaneously!
So, in place of the term "birth certificate," please read: "form that was issued at the time of the candidate's birth, which would have contained more information than the post-2001 short-form 'Certification of Live Birth' that was released by the campaign."
Of course, one doesn't necessarily have to be born in a state to get a Certification of Live Birth from that state, but I do believe that he was born in Hawaii...and my point was that it was likely other information on the form that he doesn't want to release.