You are wrong, and Jill Pryor is also wrong. Here is a rebuttal from 1916. Here is a rebuttal from 1884. Either is a more accurate essay than what Pryor wrote.
If the U.S. Congress had never passed any laws regarding U.S. citizenship at birth, you could well be correct, since it is certainly arguable that, at common law (which applies when there are no statutes on point), citizens at birth (i.e., natural-born citizens) were limited to persons who both were born within the geographic territory of the nation and had parents who were citizens of the nation. The unanimous opinion of the U.S. Supreme Court in the case of Minor v. Happersett, 88 U.S. 162 (1874), makes this point, although it doesn’t answer with certainty (because it was unnecessary for the adjudication of the case before the Court) whether both conditions were necessary for natural-born citizenship under common law: “The Constitution does not, in words, say who shall be natural-born citizens. Resort must be had elsewhere to ascertain that. At common-law, with the nomenclature of which the framers of the Constitution were familiar, it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also. These were natives, or natural-born citizens, as distinguished from aliens or foreigners. Some authorities go further and include as citizens children born within the jurisdiction without reference to the citizenship of their parents. As to this class there have been doubts, but never as to the first. For the purposes of this case it is not necessary to solve these doubts. It is sufficient for everything we have now to consider that all children born of citizen parents within the jurisdiction are themselves citizens.” http://caselaw.lp.findlaw.com/scripts/getcase.pl?navby=case&court=us&vol=88&page=162 “Natural-born” is used by the Court in contradistinction to “naturalized,” and the first means someone who us a citizen at birth, while the latter means someone who becomes a citizen later in life.
Of course, Congress *has* legislated regarding citizenship at birth, and did so as early as 1790, as the Happersett Court noted: “Congress, as early as 1790, provided that ... the children of citizens of the United States that might be born beyond the sea, or out of the limits of the United States, should be considered as natural-born citizens.” And Happersett went on to note that in 1855 this conferrance of birthright citizenship to the foreign-born children of U.S. citizens “was somewhat extended, and all persons theretofore born or thereafter to be born out of the limits of the jurisdiction of the United States, whose fathers were, or should be at the time of their birth, citizens of the United States, were declared to be citizens also.” When Congress passes a law, such statute supercedes the common law on that particular matter, and, as the Supreme Court noted in Happersett, one does not need to wonder whether the foreign-born children of U.S. citizens would be natural-born citizens at common law, because a congressional statute specifies when such children would be considered U.S. citizens at birth. Today, of course, the U.S.-citizen parent(s) of foreign-born children must meet certain residency requirements for their foreign-born children to be deemd natural-born citizens.
Regarding children born in America to non-citizen parents, Congress legislated specifically to confer U.S. citizenship at birth to such children so long as they are “subject to the jurisdiction” of the United States; this statutory language tracks that of the first sentence of Section 1 of the 14th Amendment, and is currently codified in 8 U.S.C.S. 1401(a), and has remained unchanged since at least 1952 (and I believe the predecessor of the 1952 statute had identical language with respect to this clause). Now, reasonable minds may disagree on whether the children of illegal aliens should be covered by this clase, since “subject to the jurisdiction thereof” implies that the person is legally within the jurisdiction, and I believe that the courts have been incorrect in interpreting both this statute and the 14th Amendment as conferring birthright citizenship on the U.S.-born children of illegal aliens, tourists and other persons who were neither U.S. citizens not permanent residents of the U.S. But it is incontrovertible that permanent residents of the U.S. are subject to the jurisdiction of the U.S., and thus that their children are U.S. citizens at birth.
So, to sum up, given that the term “natural-born citizen” is synonymous with “citizen at birth” or “birthright citizen,” and the fact that congressional statutes supercede the common law, all persons who are U.S. citizens at birth pursuant to U.S. law are natural-born citizens of the United States. Thus, George Romney (Mitt’s father, who was born in Mexico to two U.S. citizens and who uncontroversially ran for president in 1968) and Chester Arthur (who was elected vice president in 1880 and became president upon the death of James Garfield in 1881 despite his father not having been a U.S. citizen when his son was born) were natural-born citizens of the United States and eligible for the presidency. And the same is true for John McCain and Marco Rubio (and, assuming that he was born in Hawaii, Barack Obama).