Here is a paste from the 1790 Naturaliztion Act, the only other place "natural born citizen" is used.
I don't think there is any other US law to address Natural Born citizen. Other laws have address who becomes a citizen but not Natural born.
"In order to address one's good character, the law required two years of residence in the United States and one year in the state of residence, prior to applying for citizenship. When those requirements were met, an immigrant could file a Petition for Naturalization with "any common law court of record" having jurisdiction over his residence. Once convinced of the applicant’s good moral character, the court would administer an oath of allegiance to support the
Constitution of the United States. The clerk of the court was to make a record of these proceedings, and "thereupon such person shall be considered as a citizen of the United States."
The Act also establishes the United States citizenship of certain children of citizens, born abroad, without the need for naturalization: "the children of citizens of the United States that may be born beyond Sea, or out of the limits of the United States, shall be considered as natural born Citizens:
provided, That the right of citizenship shall not descend to persons whose fathers have never been resident in the United States".