That is completely incorrect. Obama released his birth certificate in June 2008, while still a candidate, and in fact was the first presidential candidate in US history ever to release his actual birth certificate. No sitting president ever had released their birth certificate before. Some did afterward, but never while in office. Obama set a precedent by doing so.

What he did two years later was release the long form, for which he had to receive special permission from the state of Hawaii. But he had already released his official certified birth certificate in 2008 before the election.