No, they went on to deny Vattel’s definition and use the one provided by English common law.
You’re confused as to the purpose of nations and laws, Mr. Rogers. We were never bound by the laws of England, common or otherwise, once the new nation was established. Founder George Mason walked out on the Convention and refused to sign because, in his words, the common law was completely overturned, meaning English common law. And it was.
Mason’s insistence upon a Bill Of Rights, which was as you’ll recall the result of Constitutional Amendment immediately following ratification, is the sole extent of the old common law in United States Constitutional law.
The rest drew upon numerous sources, from the Roman Republic onward, much as the Law Of Nations did. You seem fixated upon Vattel as if this somehow diminishes or discredits the Law Of Nations. He merely published a treatise upon it, he didn’t create it. He was, however, tremendously influential during the timeframe in question for having done so, far more among our Founders than any expert upon the English feudal legal system would have been.
That system is what they were removing themselves from. Why is this so difficult for you?