How is Vattel’s definition at the “core” of a case which has nothing to do with presidential eligibility? It does not seem to follow from what the case is about.
In the course of the arguments, the United States argued that while Ark was born in the United States, he was not a “natural born citizen”.
The court rejected this in it’s statement:
“The Fourteenth Amendment of the Constitution, in the declaration that
all persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside,
contemplates two sources of citizenship, and two only: birth and naturalization.”
The problem with the arguments all argued on this thread and on this site is that they are all argued persuasively in the dissent in the Ark case.
https://www.law.cornell.edu/supremecourt/text/169/649#writing-USSC_CR_0169_0649_ZO
Please go to this link and read the dissent in the Ark case. It uses the exact same arguments.