The phrase "natural-born" was coined by Vattel, and Vattel's works were in print and well-known years before the time of the writing of the Constitution.
John Jay (who became the first Supreme Court Justice) wrote George Washington concerning Jay's fears over the need for the unquestioned loyalty to our nation, and our nation alone, of any future Presidents. For this reason, Jay stated the need that future prospective Presidents be "natural-born citizens": those who could claim *both* jus sanguinis *and* jus soli, a requirement unique to only Presidents and Vice-Presidents.
Jay wanted not just jus sanguinis... (citizen by blood, both parents being citizens, which was enough to grant natural-born status to common citizens, no matter where they were born)... but jus soli (citizen by soil, born on U.S. soil) as well, a double-proof citizenship that insured a status of "natural-born citizen", not just "citizen".
This was a unique requirement specifically and only for any prospective President or Vice-President. The Naturalization Act is nonapplicable to this unique designation for the purposes of determining the eligibility of those seeking the office of President or Vice-President.
That's why it was necessary, in 2008 S. Res. 511, to clarify the Presidential eligibility of McCain and others like him who were born to military parents, both of whom were citizens, but stationed during military service on foreign soil at the time of the prospective President's birth. If not, McCain, and any others like him born of citizen parents while serving on foreign soil, would not have met the requirements of jus soli. Such an ammendment would not have been needed if the Naturalization Act definition of "natural-born citizen" applied to potential United States Presidents.
Interestingly, Obama was a sponsor of that bill. In the bill it states:
Whereas such limitations would be inconsistent with the purpose and intent of the `natural born Citizen' clause of the Constitution of the United States, as evidenced by the First Congress's own statute defining the term `natural born Citizen';
Therefore, no matter jus soli, which is in hot debate, Obama fails on the first, jus sanguinis, because of his father's alien status as a British subject... and Obama knows it.
Both Franklin and Washington were familiar with Vattel. See:
(I have archived a copy of the above page.)
As a further proof that the Framers of the Constitution used Vattel's Law of Nations we need look no further than the Constitution itself, which states under Section 8 -- Powers of Congress:
To define and punish Piracies and Felonies committed on the high Seas, and Offenses against the Law of Nations;