I think one of the reasons the distinction has been lost is because of the Grandfather clause. It was something like 40 years before we had a natural born citizen President, and so for forty years, the topic simply didn't get discussed.
By the time those forty years had passed, the English Common law argument had been heard so many times, people simply accepted it as the basis of our "natural citizen" requirement. (Not all. Pennsylvania specifically noted that we followed the Vattel definition, rather than English Common law. )
I will further point out that the Abolitionist movement gained a great deal of strength after the Revolution era. Their efforts to emancipate the slaves REQUIRED the usage of English common law, because the "Law of Nations" was very much against them.
I think they simply pushed for the acceptance of the English law definition, not because it was accurate, but because that was what they needed to obtain their goals.
Historians have agreed throughout US history that all of our early Presidents were natural born citizens; that the grandfather clause was not for the sake of George Washington, John Adams, Thomas Jefferson, James Madison, James Monroe, etc., but that it was for the likes of Alexander Hamilton (born on the island of Nevis in the Caribbean) and James Wilson (born in Scotland) that the grandfather clause was passed.
No serious historian has ever claimed otherwise.
Here is some documentation that establishes that fact.
“It was doubtless introduced (for it has now become by lapse of time merely nominal, and will soon become wholly extinct) out of respect to those distinguished revolutionary patriots, who were born in a foreign land, and yet had entitled themselves to high honours in their adopted country….” United States Supreme Court Justice Joseph Story, Commentaries on the Constitution (1833)
“The exception as to those who were citizens at the time of the adoption of the Constitution, was justly due to those men who had united themselves with the fate of the new nation, and rendered eminent services in achieving its independence ; and is, necessarily, of limited continuance.” James Bayard, A brief exposition of the Constitution of the United States, pg. 96 (1833)
“Why was this exception then made ? From gratitude to those distinguished foreigners who had taken part with us during the Revolution.” John Seely Hart, A Brief Exposition of the Constitution of the United States, pg. 71 (1860)
“The idea then arose that no number of years could properly prepare a foreigner for the office of president; but as men of other lands had spilled their blood in the cause of the United States, and had assisted at every stage of the formation of their institutions, the committee of states who were charged with all unfinished business proposed, on the fourth of September, that ” no person except a natural-born citizen, or a citizen of the United States at the tune of the adoption of this constitution, should be eligible to the office of president.” George Bancroft, History of the formation of the Constitution of the United States of America pg 346 (1866)
“The exception in favor of such persons of foreign birth as were citizens of the United States at the time of the adoption of the Constitution, is now practically extinct. The distinguished patriots who had so faithfully served their adopted country during the revolutionary struggle, and out of respect and gratitude to whom this exception was introduced into the Constitution, have all passed away. No one, therefore, but a natural born citizen can now be elected to the office of President.” Henry Flanders, An Exposition of the Constitution of the United States (1877)
“The exception to the “natural born” qualification was the Convention’s way of paying an extraordinary compliment to Alexander Hamilton and James Wilson, two distinguished members of the Convention who were foreign born. Of course, any other foreign- born citizen having the other qualifications would have been eligible, but the clause was drawn in favor of the two statesmen here mentioned.” Edward Waterman Townsend, Our Constitution: Why and how it was Made – who Made It, and what it is pg 186 (1906)
This understanding is also confirmed by James Madison's reasoning in the Smith case. He said, in essence, that Smith had been a citizen of the community that would become the State of South Carolina since his very birth:
"I conceive the colonies remained as a political society, detached from their former connection with another society, without dissolving into a state of nature; but capable of substituting a new form of government in the place of the old one, which they had for special considerations abolished. Suppose the state of South Carolina should think proper to revise her constitution, abolish that which now exists, and establish another form of government: Surely this would not dissolve the social compact. It would not throw them back into a state of nature. It would not dissolve the union between the individual members of that society. It would leave them in perfect society, changing only the mode of action, which they are always at liberty to arrange. Mr. Smith being then, at the declaration of independence, a minor, but being a member of that particular society, he became, in my opinion, bound by the decision of the society with respect to the question of independence and change of government; and if afterward he had taken part with the enemies of his country, he would have been guilty of treason against that government to which he owed allegiance, and would have been liable to be prosecuted as a traitor."