Yep. "Citizens." Not "natural-born citizens." Only those born in the U.S. to U.S. citizen parents are defined as "natural-born citizens."
And citing quotes referring to natural-born subjects is irrelevant. The U.S. has citizens, not subjects. The U.S. is a Constitutional Republic, not a Constitutional Monarchy. The U.S. fought a revolution and declared independence from the Monarchy. Remember?
Since the adoption of the 14th Amendment, the Supreme Court has recognized only two classes of U.S. Citizenship and Congress has passed no legislation altering or further clarifying the 14th Amendment’s citizenship clause.
Class One: Citizens of the United States At Birth/Natural Born Citizens.
Class Two: Naturalized United States Citizens
Class One can be president or vice president
Class Two: Cannot be president or vice president
Elk v Wilkins, 112 U. S. 94 (1884):
“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.
[The citizenship clause] “section [of the 14th Amendment] 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.”