“Citizen by birth and natural born citizen are one and the same. “
You can’t reasonably infer that from Wong Kim Ark, not least of which is because they discussed natural born citizen at length, and deliberately decided not to rule on the presidential qualifications. Wong Kim Ark cannot erase the words “natural born citizen” from the Constitution.
You most certainly can. In referring to 'natural-born citizen', Justice Gray writes, "The Constitution nowhere defines the meaning of these words, either by way of inclusion or of exclusion, except insofar as this is done by the affirmative declaration that "all persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States." In this as in other respects, it must be interpreted in the light of the common law, the principles and history of which were familiarly known to the framers of the Constitution." He goes on to detail instance after instance where English common law clearly states that persons born in the country are natural-born citizens regardless of the nationality of their parents. Justice Gray also points out that "The Fourteenth Amendment of the Constitution...contemplates two sources of citizenship, and two only: birth and naturalization. Citizenship by naturalization can only be acquired by naturalization under the authority and in the forms of law. But citizenship by birth is established by the mere fact of birth under the circumstances defined in the Constitution. Every person born in the United States, and subject to the jurisdiction thereof, becomes at once a citizen of the United States, and needs no naturalization."
The whole Ark decision can be summed up in that. Two classes of citizenship, not three. Children born in the U.S. are citizens at birth and need no natrualization. Nationality of the parents are, in most cases, irrelevant to the nationality of the child if born in the U.S. Under all this Obama is clearly qualified for the presidency if he was born in Hawaii.