The only place Natural Born Citizen is defined is in Vattel's law of Nation
Vattel's Law of Nation 212 Book 1
"Citizens and natives.
The citizens are the members of the civil society; bound to this society by certain duties, and subject to its authority,
they equally participate in its advantages. The natives, or natural-born citizens, are those born in the country,
of parents who are citizens. As the society cannot exist and perpetuate itself otherwise than by the children of the citizens,
those children naturally follow the condition of their fathers, and succeed to all their rights. The society is supposed to desire this,
in consequence of what it owes to its own preservation; and it is presumed, as matter of course, that each citizen, on entering into society,
reserves to his children the right of becoming members of it. The country of the fathers is therefore that of the children;
and these become true citizens merely by their tacit consent. We shall soon see whether, on their coming to the years of discretion,
they may renounce their right, and what they owe to the society in which they were born. I say, that, in order to be of the country,
it is necessary that a person be born of a father who is a citizen; for, if he is born there of a foreigner,
it will be only the place of his birth, and not his country."
Then the Naturalization act of 1790
Also confirms this line of thinking but in the other way, you can still be a Natural Born Citizen when born outside of the US.
" Act of March 26, 1790 (1 Stat 103-104) (Excerpts) That any alien, being a free white person, who shall have resided within the
limits and under the jurisdiction of the United States for the term of two years, may be admitted to become a citizen thereof,
on application to any common law court of record, in any one of the States wherein he shall have resided for the term of one year at least,
and making proof to the satisfaction of such court, that he is a person of good character, and taking the oath or affirmation prescribed by law,
to support the Constitution of the United States, which oath or affirmation such court shall administer; and the clerk of such court shall record such application,
and the proceedings thereon; and thereupon such person shall be considered as a citizen of the United States. And the children of such persons so naturalized,
dwelling within the United States, being under the age of twenty-one years at the time of such naturalization,
shall also be considered as citizens of the United States. And the children of citizens of the United States, that may be born beyond sea,
or out of the limits of the United States, shall be considered as natural born citizens: Provided, that the right of citizenship shall not
descend to persons whose fathers have never been resident in the United States: . . ."
Then the Fourteen amendment redefined who was a Citizen, however, it did not go into the subject of Natural Born Citizen.
Barrack Obama's birth Certificate say he was born in Hawaii, it also lists his father as "African".
So his birth Certificate proves his place of birth was Hawaii and that his father was African.
It follows then that Barrack is a Citizen but he is not a Natural Born Citizen because his father was African and according to British law of that day, he is a British Citizen by right of descent (a Natural Born Citizen of Kenya (British)).