Its addressed there.
So far as I know the phrase “natural-born citizen” does not appear at all in US statutes.
It does however appear in this Supreme Court decision:
1875 opinion of Supreme Court Chief Justice Waite in Minor v. Happersett
In his majority opinion stated: “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.” Minor v. Happersett, 88 U.S. 162 (1875).
It seems to me the matter of Presidential eligibility is still legally undecided. It simply has never been explicitly addressed by Congress or the Supreme Court. SCOTUS had the chance in January 2009, and Chief Justice John Roberts reportedly was the missing 4th vote to hear the Obama case.
Its addressed there.
The ruling you cite, in clear English, shows that ‘natural-born’ and citizen from birth are the same thing ...
“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, ...”
So those who are citizens at birth are ‘natural-born citizens’.
The ruling also mentioned the two ways of acquiring citizenship, one being natural-born and the other naturalization. Thus the ruling comports with the simple and clear understanding that ‘natural-born citizen’ and citizen from birth are the same thing.
The idea that there is some mysterious ‘natural-born citizen’ class distinct from those who have been US citizens their entire lives but are not NBC is a bogus one that has zero basis in US law.
Cite a credible source for this or retract it. Otherwise you're just smearing three distinguished members of the court without evidence (Alito, Scalia and Thomas) by implying that they decided to hear this case.