The term "native-born" -- I don't know.
With regard to your other question, until the Cable Act of 1922, and citizenship laws of the 1930s a woman who married a man acquired his citizenship status upon marriage, not vice versa. So yes, the citizenship of the father was automatically the citizenship of the child. The citizenship of the mother was almost irrelevant because that citizenship followed that of her husband as well.
So it doesn't have to explicitly say "both parents". Anyone familiar with the law throughout American history would know that the father's citizenship was all that was important, because it was not possible for the mother to have a citizenship different from her husband.
So then, you are claiming that if Obama's mother was a citizen of Kenya and his father a US citizen, there would be no problem?