You might want to read the pinged post, it addresses your repeated mistaken impression of Elg and Ark.
I'm not sure exactly what you're getting at. Look at this quote he posted from the Ark decision: "Children, born in England, of such aliens, were therefore natural-born subjects. But the children, born within the realm, of foreign ambassadors, or the children of alien enemies, born during and within their hostile occupation of part of the king's dominions, were not natural-born subjects, because not born within the allegiance, the obedience, or the power, or, as would be said at this day, within the jurisdiction, of the king." If we accept that as true then that means that Obama is a natural born citizen, regardless of his father's citizenship, in that he was born in Hawaii and his father was not a diplomat of enemy alien. Right there that supports what I've been saying. They he quotes Justice Swayne: "All persons born in the allegiance of the King are natural-born subjects, and all persons born in the allegiance of the United States are natural-born citizens. Birth and allegiance go together." Again, supports what I've been saying.
In the Elg case, only one parent is clearly identified as having U.S. citizenship. The mother's citizenship is never stated. Elg has natural born citizenship because she was born here, regardless of parent's status.
So where, exactly, is my impression of the Elg and Ark cases incorrect?