It appears to me that upon principles of public law as well as of the common law, he must if born a British subject, be deemed to adhere to, and retain the national allegiance of his parents at the time of the treaty. Vattel considers the general doctrine to be that children generally acquire the national character of their parents, Vattel, B. 1, ch. 19. sec. 212, 219, and it is certain, both by the common law and the statute law of England, that the demandant would be deemed a British subject.
...still his national character was derivative from his parents, and was under the peculiar circumstances of this case, liable to be changed during the Revolutionary War, and that if his parents reverted to their original character as British subjects and adhered to the British Crown, his allegiance was finally fixed with theirs by the treaty of peace.
During the civil war (Revolution against Britain), the disputed territories had mixed citizenship. This was an exceptional situation, where people living side by side could be citizens of different countries.
That differed from the norm, where in both Great Britain & the USA, birth determined citizenship. There is no doubt that someone born in Britain, even if there temporarily, was a natural born subject. In like manner, someone born in the USA, even if their parents were there temporarily (but legally), is a natural born citizen.
The intent and desire of the parents applies when you have territory in dispute during civil war. But there is a long body of legal decisions backing the idea that anyone born in the USA is a natural born citizen. WKA reviewed dozens of cases that decided that.
What birthers have been unable to find is a single case where it was ruled that a free person born in the USA needed naturalization. There is not a single case where the courts ruled a free person born in the USA was not a citizen by birth, apart from the disputed territories during the Revolutionary War.
Children who are dual citizens have the right to choose which country they will be citizens of. They are assumed to follow their parents until they are old enough to decide for themselves. Thus Elg, after living for years in Sweden as a Swedish citizen (following her parents) was ruled a natural born citizen of the US by birth, choosing to follow it when she was 18:
“But the Secretary of State, according to the allegation of the bill of complaint, had refused to issue a passport to Miss Elg “solely on the ground that she had lost her native born American citizenship.” The court below, properly recognizing the existence of an actual controversy with the defendants (Aetna Life Ins. Co. v. Haworth, 300 U. S. 227), declared Miss Elg “to be a natural born citizen of the United States,” and we think that the decree should include the Secretary of State as well as the other defendants. The decree in that sense would in no way interfere with the exercise of the Secretary’s discretion with respect to the issue of a passport, but would simply preclude the denial of a passport on the sole ground that Miss Elg had lost her American citizenship.”
http://supreme.justia.com/us/307/325/case.html