There is of course no such law. But there is also not any law that says they are the same thing.
But it's easy to prove that "citizen at birth" and "native born citizen" are not the same thing. Persons born outside the US of two US citizen parents are citizens at birth, but they are not "native born". And in fact, the Supreme Court has ruled that, with some eceptions, persons born outside of the US, but citizens at birth under the statutes are "naturalized at birth" and are not even 14th amendment citizens (born or naturalized in the US), even though they do not need to go through the usual statutory naturalization process. The same was true of residents of lands acquired by the US, such as Puerto Rico, they were naturalized en mass, and did not need to go through the naturalization process.
But as to "Natural Born Citizen", the issue can only arrise, in cases involving Presidential or Vice Presidential eligibility. There are no other circumstances where being a "natural born citizen" matters. Native born may matter, "naturalized in the US" verses naturalized outside the US may matter, and boht have been the subjects of many cases. But no cases exist at the Supreme Court level at least, where "natural born citizen" matters.
Here is a link from Justia, with links to applicable case law, that states:
The first sentence of � 1 of the Fourteenth Amendment contemplates two sources of citizenship and two only: birth and naturalization.1230 This contemplation is given statutory expression in � 301 of the Immigration and Nationality Act of 1952,1231 which itemizes those categories of persons who are citizens of the United States at birth; all other persons in order to become citizens must pass through the naturalization process. The first category merely tracks the language of the first sentence of � 1 of the Fourteenth Amendment in declaring that all persons born in the United States and subject to the jurisdiction thereof are citizens by birth.1232 But there are six other categories of citizens by birth. They are: (2) a person born in the United States to a member of an Indian, Eskimo, Aleutian, or other aboriginal tribe, (3) a person born outside the United States of citizen parents one of whom has been resident in the United States, (4) a person born outside the United States of one citizen parent who has been continuously resident in the United States for one year prior to the birth and of a parent who is a national but not a citizen, (5) a person born in an outlying possession of the United States of one citizen parent who has been continuously resident in the United States or an outlying possession for one year prior to the birth, (6) a person of unknown parentage found in the United States while under the age of five unless prior to his twenty-first birthday he is shown not to have been born in the United States, and (7) a person born outside the United States of an alien parent and a citizen parent who has been resident in the United States for a period of ten years, provided the person is to lose his citizenship unless he resides continuously in the United States for a period of five years between his fourteenth and twenty-eighth birthdays.
So right there you have 7 types of "citzen at birth". Most of which are clearly not "native born".
I appreciate your research to verify that ALL citizens-at-birth qualify as “natural born citizens.”
Its really very, very simple: if you are an American citizen but not a naturalized citizen and therefore ineligible to be president or vice president then you are a citizen-at-birth and you qualify to be president or vice president as a natural born citizen.
A person who was in one of those seven categories of citizens-at-birth that you named was Charles Curtis, the 31st Vice President of the United States who was the first person with acknowledged Native American ancestry to reach either of the two highest offices in the United States government’s executive branch. His maternal ancestry was three-quarters’ Native American, of ethnic Kaw, Osage and Pottawatomie ancestry. Curtis spent years of his childhood living with his maternal grandparents on the Kaw reservation.
The fact that Barack Obama is President of the United States and that John McCain COULD have been president of the United States is verification of the above.
The national Republican Party, the McCain-Palin campaign and all of the Republican attorneys in the House of Representatives and the US Senate would have never allowed Obama’s elevation to the presidency without a legal challenge if what I have stated was untrue. Nor would the US Supreme Court have rejected all Obama eligibility challenges from being heard if there were constitutional grounds to disqualify him.