The law simply isn't clear on this point. While the SCOTUS seemed to treat "natural-born citizenship" and "citizenship" interchangeably in Wong Kim Ark, the counter-argument to that is that that was a case about a child's inheritance, not about who is eligible to be President of the United States. IOW, it's likely the Court would have to accept an argument that, at least as to the President, the Founders intended the Constitution to require something beyond citizenship acquired merely by place of birth.
It's not impossible to make or have such an argument accepted. There is particular evidence that the Founders did not want those of dual citizenship, or -- based on their parents' citizenship -- divided allegiance, installed in the presidency. And so on. But the bottom line is that the Supreme Court has never actually reviewed the issue of who is a "natural-born citizen" in the context of what the Founders were intending as to the presidency.
The decision in Wong Kim Ark was careful not to call the plaintiff a natural born citizen, plus it emphasizes that the child was considered a citizen by virtue of the 14th amendment in part because the parents were permanent U.S. residents. This was not the case for Obama’s father. Wong Kim Ark also echoed Minor V. Happersett’s use of the Vattel definition of natural born citizen: born of citizen parents on native soil. That’s two strikes against Obama.