Mark Lilla, writing in the New York Times, nailed how the key Founders understood religion.
In truth, the leaders of the British and American Enlightenments shared the same hope as the French lumières: that the centuries-old struggle between church and state could be brought to an end, and along with it the fanaticism, superstition and obscurantism into which Christian culture had sunk. What distinguished thinkers like David Hume and John Adams from their French counterparts was not their ultimate aims; it was their understanding of religious psychology. The British and Americans made two wagers. The first was that religious sects, if they were guaranteed liberty, would grow attached to liberal democracy and obey its norms. The second was that entering the public square would liberalize them doctrinally, that they would become less credulous and dogmatic, more sober and rational.
In other words, the Founders desired the Christian religion to reform and conform to the tenets of the age of Enlightenment. One major difficulty with the Founders' vision is most of the "irrational" tenets of the Christian religion which they desired Christians to scrap turned out to be the heart and soul of orthodox Christianity: The Trinity, the Incarnation, the Atonement, the infallibility of Scripture, etc. While Founders like Adams and Jefferson, and the preachers they followed like Jonathan Mayhew and Charles Chauncy, continued to call themselves "Christian," they sacrificed so many of the key tenets of Christianity at the alter of rationalism (as Gregg Frazer puts it), that their religious creed arguably ceased to be "Christian." This is why Dr. Frazer coins a new term to describe their faith: Theistic Rationalism.
See Dr. Frazer painstakingly debate this issue on these threads with commenter Richard Knapton, who for some reason, seems intent on denying the historical reality that the Founders' desire that the Christian religion further reform was part of their "Enlightenment project."
Since I have quoted extensively from Jefferson and Adams to prove the point in the past, let me, for a change, offer some quotations from Washington and Madison, to demonstrate that our Founders' personal religious beliefs and desire for religious reform was part of their Enlightenment project.
First, James Madison. From this excellent paper by James H. Hutson on Madison's religion. Hutson writes:
Perhaps a better clue to Madison's outlook is a letter to Jefferson, December 31, 1824, in which he complained about Presbyterian "Sectarian Seminaries," armed with charters of incorporation, disseminating obsolete religious doctrines, by which he clearly meant Calvinism.
Unassailable charters allowed a "creed however absurd or contrary to that of a more enlightened Age" [Rowe's emphasis] to be perpetuated indefinitely. The Reformation itself, Madison continued, must be considered the "greatest of abuses," if legal impediments could prevent its doctrines from being brought up to date. The idea that Madison was espousing, that religious truth must evolve to incorporate the discoveries of science and other branches of modern learning, was far from the theological orthodoxy of most 19th century American churches. It can be inferred that his own religious views had evolved from the verities he had learned at Princeton, but how and in what direction neither this nor other writings disclose.
Religious controversies are always productive of more acrimony and irreconcilable hatreds than those which spring from any other cause. I had hoped that liberal and enlightened thought [Rowe's emphasis] would have reconciled the Christians so that their religious fights would not endanger the peace of Society.
Letter to Sir Edward Newenham, June 22, 1792
Also noticed how Washington referred to Christians in the third person as "them" or "their" as though he were not part of that group. Indeed, unlike Adams and Jefferson, Washington didn't even call himself a Christian.
The term "liberal" goes hand in hand with "enlightened" because our classical liberal founding, though it drew from a variety of different intellectual sources, was predominantly a product of Enlightenment thought.
Now, understanding the context, look at John Adams's thoughts on the Jews and his desires for them. Each of the bold words is my emphasis:
I really wish the Jews again in Judea an independent nation. For as I believe the most enlightened men of it have participated in the amelioration of the philosophy of the age, once restored to an independent government & no longer persecuted they would soon wear away some of the asperities and peculiarities of their character [and] possibly in time become liberal unitarian Christians for your Jehovah is our Jehovah & your God of Abraham Isaac & Jacob is our God.
John Adams to Mordecai Noah, March 15, 1819. Adams Papers (microfilm), reel 123. Quoted from James H. Hutson's The Founders on Religion, p. 127.
The ameliorated "philosophy of the age" was Enlightenment, whose "enlightened men," (of which he, of course, was one) reformed Christianity to become in Adams's words "liberal unitarian." The problem again, is this "enlightened" version of Christianity which rejects the Trinity, Eternal Damnation, infallibility of Scripture, etc. is not "Chrisitanity" as the orthodox believers understand that term.