No, it's not. Chief Justice Waite did not cite de Vattel at all. His writings appear nowhere in the Court's unanimous opinion.
The "nomenclature" referenced would therefore be something other than English, since Waite's description of the particulars of natural born citizenship do not correspond to English common law.
Which would be, what, exactly?
I could attempt to paraphrase this several more ways, if that would help. I'm accustomed to doing so when dealing with offshore suppliers in China and Korea. They find it helpful. Do you?
Good to know that you're a helpful fellow with your Asian trading partners, but paraphrasing won't be necessary here. Pointing out what "something other than English" is, however, would be great.