(1) Vattel was born in Switzerland, was a counselor (ambassador) for the Kingdom of Saxony, and wrote “Le Droit des Gens” (The Law of Nations) in the French language.
(2) The work was translated into English, published in London in 1759 and went through numerous editions, including American editions of 1796, 1833 and 1852, the 1883 reprint of the 1852 edition being available on the internet. Vattel’s book was the most influential work on international and constitutional law in the period 1758-1900.
(3) Ben Franklin wrote to Dutch diplomat Charles Dumas in 1775, “I am much obliged by the kind present you have made us of your edition of Vattel. It came to us in good season, when the circumstances of a rising state make it necessary frequently to consult the law of nations. Accordingly, that copy which I kept, (after depositing one in our public library here, and sending the other to the College of Massachusetts Bay, as you directed,) has been continually in the hands of the members of our Congress, now sitting, who are much pleased with your notes and preface, and have entertained a high and just esteem for their author.”
(4) John Adams thought so highly of Vattel that he considered him a father-figure, writing in his diary for February 1, 1763 after wasting his time, “I employed however, too little of my Time in Reading and in Thinking. I might have spent much more. The Idea of M. de Vattell indeed, [illegible] scowling and frowning, haunted me.”
(5) James Madison, in his instructions to John Jay in Madrid on October 17, 1780, quoted Vattel concerning American rights to sail on the Mississippi, “An innocent passage (says Vattel) is due to all nations with whom a State is at peace.”
(6) Delegates to the Continental Congresses met at Carpenters Hall, which was the building that also housed the Library Company of Philadelphia. The librarian reported that Vattel was the primary source read by the delegates during the First Continental Congress. The Founding Fathers were so familiar with the works of Vattel that nobody felt the need to define natural-born citizen, since Vattel had already done so.
(7) The phrase “the law of nations” appears in Article 1 of the U.S. Constitution, “To define and punish piracies and felonies committed on the high seas, and offenses against the law of nations.”
Which to use when there is a conflict is the question. Many of the authors of the Constitution, and many of those who ratified it, were lawyers, and would have been familiar with both, but especially "..Laws of England", since until at 1775, they were English lawyers. But I'm still not sure which to use in this case.
There is the additional factor of the 14th amendment which does define persons born in the United states, and subject to their jurisdiction (which children of ambassadors, members of conquering armies and so forth, are not) as citizens.
But still I could be convinced, or convince myself either way. It's a matter for the courts, and they need to get on with it.
Of course if they do, it's possible, even likely, that they will find that the point is moot, as the presumptive President-elect may turn out not to be a natural born citizen under either English Common law or "Laws of Nations" definitions.