I've heard Bill Clinton was an ESFP...and that actually seems more accurate to me, considering the whole Monica thing.
Obama is difficult. He's a very private man, but he goes out to where the people are. His campaign manager was/is a genius, and it feels like Obama hand selected him. Behind the scenes rumor is Obama is a stern leader who is sometimes harsh when he doesn't see the results he wants.
I think Obama could either be NT or NF - it's difficult to say because we haven't seen him in action yet. But he's very calculating and precise. Though he's young, he knows exactly what's needed to get people working...and he knows exactly the right words to say to get people motivated and listening. He's definitely intuitive. But he could be an INFJ if his main reason for getting into politics is as he said: To institute change. Heck, I'd get into politics if I felt I could seriously make a difference.
I'd probably get out of it once it got too hot and uncomfortable and my introversion was tested, but if an INFJ seriously feels this is the way to make a true difference, I could see them as President. Despite how uncomfortable it made them.
ETA: But they'd also get seriously, seriously hurt because they couldn't imagine anyone betraying them.