Obama has birthright citizenship because his mother was a US citizen. And because he was born on US soil. But his father was a British subject, so he's not a natural born citizen as required by Article II, and as defined by the body of international law referred to in the Constitution as the Law of Nations, whose preeminent codification at the time of the Constitutional Convention was in a book by de Vattel.
Minor v. Happersett , 88 U.S. 162 (1875):
The Constitution does not in words say who shall be natural-born citizens. Resort must be had elsewhere to ascertain that. At common law, with the nomenclature of which the framers of the Constitution were familiar, it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also. These were natives or natural-born citizens, as distinguished from aliens or foreigners.
Correspondence among the Framers makes it clear that the issue that motivated the "natural born citizen" requirement was to ensure that the Commander-in-chief of the military would have absolutely no allegiances or ties to any foreign power, and would not be a citizen or subject of any foreign power, nor be able to make any such claim. Obama's father was a British subject at the time of his birth, and Obama can legally claim to be a British subject based on that fact.
That case had to do with voting rights in Missouri.
The quote from the decision in that case does not supercede Title 8 USC Sec 1401.