In 1795 the Naturalization Act from 1790 was changed and children born abroad to US citizen parents were considered as citizens (at birth). It is obvious that early legislators made distinction between the phrases: “citizen at birth” vs. “natural-born citizen”.
Why would they change the language of the law if two phrases had the same meaning?
Equivalent language is used all the time. And someone born abroad with 2 US citizen parents IS considered a NBC - see John McCain.