It's the facts you ignore that are more telling. Fact: John Adams was not part of the Constitutional convention. Fact: John Adams wrote no known letters to any member of the convention discussing natural-born citizenship. Fact: Nothing in the Adams papers on file with the Massachusetts Historical Society show that Adams ever corresponded with any of the Founders on the definition of natural-born citizen. Fact: In his book you referenced, "Defense of the Constitutions of the United States", Adams doesn't mention Vattel once that I can find. Fact: The Adams Papers with the Massachusetts Historical Society contain over 27,000 separate items. The idea that there is an unknown cache of John Adams papers somewhere is ridiculous.
Birther.