Did you miss this right before your snippet?
And what of this from after your snippet? It's really important as it makes a critical distinction.
You have to read this stuff in context as you well know.
Nothing makes these people natural born citizens, just citizens.
An alien visiting on a student visa retains his, or her, citizenship due to a dependent power...their home nation's claim to its own citizens.
Your snippet even emphasizes this...
You fail to distinguish between Indians, whose nations are also located totally inside the USA, and aliens. The law understood that distinction:
“An Indian, born a member of one of the Indian tribes within the United States, which still exists and is recognized as a tribe by the government of the United States, who has voluntarily separated himself from his tribe, and taken up his residence among the white citizens of a state, but who has not been naturalized, or taxed, or recognized as a citizen either by the United States or by the state, is not a citizen of the United States within the meaning of the first section of the Fourteenth Article of Amendment of the Constitution.”
However, that changed by law in the 1900s.
WKA also noted that exception:
“The only adjudication that has been made by this court upon the meaning of the clause, “and subject to the jurisdiction thereof,” in the leading provision of the Fourteenth Amendment is Elk v. Wilkins, 112 U.S. 94, in which it was decided that an Indian born a member of one of the Indian tribes within the United States, which still existed and was recognized as an Indian tribe by the United States, who had voluntarily separated himself from his tribe and taken up his residence among the white citizens of a State but who did not appear to have been naturalized, or taxed, or in any way recognized or treated as a citizen either by the United States or by the State, was not a citizen of the United States, as a “person born in the United States and subject to the jurisdiction thereof” within the meaning of the clause in question.
That decision was placed upon the grounds that the meaning of those words was
not merely subject in some respect or degree to the jurisdiction of the United States, but completely subject to their political jurisdiction, and owing them direct and immediate allegiance;
that, by the Constitution, as originally established, “Indians not taxed” were excluded from the persons according to whose numbers representatives in Congress and direct taxes were apportioned among the [p681] several States, and Congress was empowered to regulate commerce not only “with foreign nations” and among the several States, but “with the Indian tribes;” that the Indian tribes, being within the territorial limits of the United States, were not, strictly speaking, foreign States, but were alien nations, distinct political communities, the members of which owed immediate allegiance to their several tribes and were not part of the people of the United States; that the alien and dependent condition of the members of one of those tribes could not be put off at their own will without the action or assent of the United States, and that they were never deemed citizens except when naturalized, collectively or individually, under explicit provisions of a treaty, or of an act of Congress; and therefore that
Indians born within the territorial limits of the United States, members of, and owing immediate allegiance to, one of the Indian tribes (an alien, though dependent, power), although in a geographical sense born in the United States, are no more “born in the United States, and subject to the jurisdiction thereof” within the meaning of the first section of the Fourteenth Amendment than the children of subjects of any foreign government born within the domain of that government, or the children born within the United States of ambassadors or other public ministers of foreign nations.
And it was observed that the language used in defining citizenship in the first section of the Civil Rights Act of 1866, by the very Congress which framed the Fourteenth Amendment, was “all persons born in the United States, and not subject to any foreign power, excluding Indians not taxed.” 112 U.S. 99-103.
Mr. Justice Harlan and Mr. Justice Woods, dissenting, were of opinion that the Indian in question, having severed himself from his tribe and become a bona fide resident of a State, had thereby become subject to the jurisdiction of the United States within the meaning of the Fourteenth Amendment; and, in reference to the Civil Rights Act of 1866, said:
Beyond question, by that act, national citizenship was conferred directly upon all persons in this country, of whatever race (excluding only “Indians not taxed”), who were born within [p682] the territorial limits of the United States, and were not subject to any foreign power.
And that view was supported by reference to the debates in the Senate upon that act, and to the ineffectual veto thereof by President Johnson in which he said:
By the first section of the bill, all persons born in the United States, and not subject to any foreign power, excluding Indians not taxed, are declared to be citizens of the United States. This provision comprehends the Chinese of the Pacific States, Indians subject to taxation, the people called Gypsies, as well as the entire race designated as blacks, persons of color, negroes, mulattoes, and persons of African blood. Every individual of those races, born in the United States, is, by the bill, made a citizen of the United States.
112 U.S. 1114.
The decision in Elk v. Wilkins concerned only members of the Indian tribes within the United States, and had no tendency to deny citizenship to children born in the United States of foreign parents of Caucasian, African or Mongolian descent not in the diplomatic service of a foreign country.
The real object of the Fourteenth Amendment of the Constitution, in qualifying the words, “All persons born in the United States” by the addition “and subject to the jurisdiction thereof,” would appear to have been to exclude, by the fewest and fittest words (besides children of members of the Indian tribes, standing in a peculiar relation to the National Government, unknown to the common law), the two classes of cases — children born of alien enemies in hostile occupation and children of diplomatic representatives of a foreign State — both of which, as has already been shown, by the law of England and by our own law from the time of the first settlement of the English colonies in America, had been recognized exceptions to the fundamental rule of citizenship by birth within the country. Calvin’s Case, 7 Rep. 1, 18b; Cockburn on Nationality, 7; Dicey Conflict of Laws, 177; Inglis v. Sailors’ Snug Harbor, 3 Pet. 99, 155; 2 Kent Com. 39, 42.
The principles upon which each of those exceptions rests were long ago distinctly stated by this court. [p683]”
http://www.law.cornell.edu/supct/html/historics/USSC_CR_0169_0649_ZO.html