The "equivalence" remark in that paragraph, you are making into "birth abroad is equivalent to birth in the US." For purposes of citizenship, this equivalence DEPENDS ON 1409. It is the statute that gives citizenship to the birth abroad case. Take the statute away, and the unwed citizen mother's child, born abroad, is an alien, period. The born abroad child is not a citizen under the constitution.
Citizen of the US and alien to (not citizen of) the US.
Looking at the US constitution, in particular the 14th amendment, the group that is citizens of the US can be subdivided into two groups.
Citizen under the constitution (born in the US) or citizen only due to operation of a statute (not born in the US). Those who are citizens solely by operation of a statute are called "naturalized citizens."
Case law is 100% uniform on this, with the only exceptions being diplomats and invaders.