Correct.
And the appeal to Vattel, as I've pointed out, is problematic as well.
Vattel's system is markedly different from ours: the children of non-citizen permanent residents are not citizens under Vattel, but are clearly US citizens under US law.
He himself points out that there are significant real-life exceptions to his ideal definitions of terms.
Problematic on various grounds, not least of which is that Vattel is not a legally binding document.