The word "natural" (and therefore the legal term of art, "natural born citizen") does not appear in 8 USC 1401. 8 USC 1401 defines "citizenship," and that terms appears as a qualifier for Congress.
-- That's why folks get upset over "anchor babies." --
Not exactly. The anchor baby phenomenon irks people because the babies are used to give immigration/naturalization preference to people who broke the immigration laws in the first place.
The qualification for the presidency is an issue that occurs infrequently, and to a small fraction of the population.
While the location of birth is highly relevant in a citizenship inquiry (countries tend to want to "control" everybody born on their turf), the citizenship of parents is ALSO highly relevant, and in some cases is absolutely determinative.
Naturalization Act of 1790 (http://pds.lib.harvard.edu/pds/viewtext/5596748?op=t&n=1)
And the children of citizens of the United States that may be born beyond fea, or out of the limits of the United States, fhall be confidered as natural born citizens: Provided, That the right of citizenfhip fhall not defcend to perfons whofe fathers have never been refident in the United States
The citizenship of the parents is only relevant when the child is born abroad. If born on US soil, it is not.
This need not justify anchor babies, however. To be a natural born citizen, a child needs to be born in the US and be "subject to the jurisdiction thereoff." That's why US-born children of diplomats aren't citizens at birth.
While a legal resident alien is clearly under US jurisdiction, it has been argued that an illegal alien, and therefore his offspring, is not. SCOTUS has not definitively resolved this matter.