Came to be regarded as the "nomenclature," how? What Court case between 1776 and 1788 put us on a different path of defining what "natural-born citizen" means than the longstanding definition of "natural-born subject" in English common law? Common law is case law, and you're not citing any cases between those years that would point to us being on a different path.
What's clear to you isn't clear at all to me, particularly since Chief Justice Waite did not cite de Vattel in the Court's unanimous opinion. Justice Gray didn't cite de Vattel either in Wong Kim Ark. (But the two dissenters in Wong Kim Ark cited de Vattel...once.)
If you want to say that by "common law," Chief Justice Waite actually meant a specific French-to-English translation of a Swiss legalist's treatise, fine. But you have to cite a CASE between 1776 and 1788 showing that we set ourselves on a different path, because common law is case law.
Besides, all of this is moot anyway since AmericanVictory pointed to the oral arguments of Tuan Anh Nguyen. Justice Scalia makes it clear that "the meaning of natural born within the Constitution...requires jus soli."
There are US legal precedents that occurred prior to Ratification under the Articles of Confederation that rather precisely underscore the points I've attempted to convey to you. However, to go to the effort to provide them to you, it is becoming increasingly evident, will be a waste of time and effort, since you are utterly incapable of recognizing that the defining qualities of a natural born citizen, as written by Chief Justice Morrison Remick "Mott" Waite, do not correspond at all to the English common law defining qualities of a natural born subject, as understood in the modern era primarily via William Blackstone's Commentaries.
It appears you are hungup on the problem of not understanding the origins of American common law.