Is Ted Cruz a naturalized citizen?

I am puzzled by the fact that some commentators seem to think that Rogers v. Bellei (1971) is not a problem for Ted Cruz.  It seems to me to be a significant problem for him. 

In Bellei, the Supreme Court held—or at least clearly affirmed—that a US citizen in Cruz’s precise circumstances (foreign birth, American mother, alien father) was a naturalized citizen.  The Court divided, 5-4, on whether the plaintiff was naturalized "in the United States" or overseas, but all nine Justices agreed that he was naturalized. 

On many occasions, the Court has stated or implied that naturalized citizens are not eligible to be president.  In Schneider v. Rusk, for example, the Court observed that “the rights of citizenship of the native born and of the naturalized person are of the same dignity, and are coextensive. The only difference drawn by the Constitution is that only the ‘natural born’ citizen is eligible to be President.”  377 U.S. 163, 165-66 (1964).

Likewise, in Luria v. United States, the Court wrote: “Under our Constitution, a naturalized citizen stands on an equal footing with the native citizen in all respects save that of eligibility to the Presidency.” 231 U.S. 9, 22 (1914).  The Court repeated this statement thirty years later in Baumgartner v. United States.  See 322 U.S. 665, 673 (1944) (quoting Luria).

In short, Cruz may be ineligible to be president for two relatively simple reasons: (1) naturalized citizens are not eligible; and (2) Cruz is a naturalized citizen. 

I am not sure yet whether I think this argument is sound, but it does seem to be an argument that Cruz and his supporters will need to address.