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There are only two types of citizens: born and naturalized. Born citizens can be president, naturalized citizens cannot.
There is no difference between a Citizen of the United States At Birth and a Natural Born Citizen.

“Citizenship is membership in a political society, and implies the reciprocal obligations as compensation for each other of a duty of allegiance on the part of the member and a duty of protection on the part of the society.
Under the Constitution of the United States, a naturalized citizen stands on an equal footing with the native citizen in all respects save that of eligibility to the Presidency.” —Luria v. United States, 231 U. S. 9 (1913)

“The distinction between citizenship by birth and citizenship by naturalization is clearly marked in the provisions of the constitution, by which ‘no person, except a natural-born citizen, or a citizen of the United States at the time of the adoption of this constitution, shall be eligible to the office of president;’ and ‘the congress shall have power to establish an uniform rule of naturalization.’Const. art. 2, § 1; art. 1, § 8.

This section contemplates two sources of citizenship, and two sources only: birth and naturalization. The persons declared to be citizens are “all persons born or naturalized in the United States, and subject to the jurisdiction thereof.”—Elk v. Wilkens, 112 U. S. 94 (1884)