Once again, the writings of 19th century Swiss philosophers are not governing documents.
And interestingly, the English version of Vattel that you cite was not published until 1797, ten years after out Constitution was ratified.
“The Law of Nations, or, principles of the law of nature, applied to the conduct and affairs of nations and sovereigns. Translated from the French by Joseph Chitty, Esq. Based on the 1797 edition, republished by T. & J. W. Johnson, law booksellers, Philadelphia (1853), together with additional notes and references by Edward D. Ingraham, Esq. Ingraham’s annotations have been omitted.”
Perhaps, our Framers had a Time Machine!
As far as what version of Vattel the framers would have access to, here it is:
Translating the French text: "les naturels, ou indigenes, sont ceux qui sont nes dans le pays, de parents citoyens"
To English, gives this: "the natural, or indigenous, are those born in the country, parents who are citizens"