The case you cite, Ludlam v. Ludlam, like so many cases cited by those who insist on a twisted interpretation of the Constitution, had nothing at all to do with a child born on US soil of non-citizen parents.
In the absence of any law of the United States governing the particular case, the question whether one born out of the United States is a citizen, is to be determined by the common law, as it existed, irrespective of English statutes, at the adoption of the Federal Constitution...
Held, accordingly, that where a citizen of the United States went to Peru at the age of eighteen years, with the intention of indefinite continuance there for the purpose of trading, but took no steps to be naturalized in Peru, or to indicate an intention of a permanent change of domicil, otherwise than as before stated, his child born to him in Peru of a wife the native of that country, is a citizen of the United States.
This case is also abundantly clear on what law its citizenship determination rested: the common law.
Once again, this case, like all of the others, is consistent with everything I've said, and everything said by every real legal authority of the early United States.