Tou are 99.9% correct
I'm not so sure, though. Ann Dunham was an American citizen at the time of Obama's birth, so even if he were born outside the United States, Obama would be a citizen from birth and, thus, would not have to be naturalized. As such, he would be eligible to become president all the same.
Correct me if I'm wrong, though.
The foreign-born child of a U.S. citizen and a non-citizen does inherit citizenship at birth, but only if certain considerations are met. These considerations are defined by statute and have changed over time.
At the time of Obama's birth, the child could inherit citizenship only if the citizen parent had lived in the United States for at least five years after the age of 14. His mother, Stanley Dunham Obama, was less than 19 years old, and therefore could not have lived in the U.S. for five years after the age of 14. This is obviously a technical flaw in the law, but by its literal meaning would have denied alt-Obama citizenship at birth.
I say it is a technical flaw, because the intent of the clause was to prevent inheritance of birth citizenship through individuals who are not "Americans", in the sense of not living in the U.S. for any significant part of their lives. If there was no such rule, children of people who had spent nearly all their lives outside the U.S. for decades could claim citizenship, and even pass it on to their children, with none of them actually living in the U.S.
But the clause would apply to Mrs. Obama, or
any parent under the age of 19, even if the citizen mother never left the U.S. until the day of the child's birth; or if the citizen father never left the U.S. at all.
If one ignores this absurd technicality, then Obama would still be eligible even if born in Kenya.