“Do people actually think Obama is not really a citizen “
Yep. Rumor has it that he was born in Kenya because his mother was trying to get to Hawaii but the airlines denied her passage as she was too pregnant. She went to Hawaii after the birth then registered him as a live birth. Anyone at any time can register a person as a live birth but that is not a birth certificate valid for citizenship.
But if he is born to a mother who is an American citizen doesn’t that mean he is an American too?