Conrad Wright is one of the sources from which Dr. Gregg Frazer's PhD thesis draws and he coined the term "supernatural rationalism" to describe what Dr. Frazer terms "theistic rationalism." This was the theology of America's key Founders and many of the divines preaching on behalf of revolution and republicanism from the pulpit.
Key to this theology was the confidence in man's reason to determine truth and elevating the findings of man's reason to be at least on par with those of biblical revelation, and often surpassing revelation as the ultimate arbiter of truth. But, unlike the deists, the "supernatural rationalists" neither categorically rejected revelation nor eschewed the label "Christian." Many of them did however, reject enough of the tenets of traditional Christianity (most notably the Trinity) that they arguably ceased being "Christian" in any meaningful sense.
This post from Transient and Permanent discusses Wright's thesis in more detail. As the post notes:
Wright calls attention to a middle way between Deism and Christian enthusiasm which appears to have been widespread. In so doing, he not only points out a common attitude but also draws the Deists and revivalistic evangelicals into sharper focus. The supernatural rationalists shared with the Deists an appreciation of reason and natural religion, while they also shared with the evangelicals a belief in the Christian revelation. For them, natural religion serves as the launching point for the special revelation of Jesus Christ, which not only doesn’t oppose reason, it is confirmed because of its accordance with reason. Christianity for them is rational, not mysterious: even miracles are basically the logical actions that an orderly God would take to intelligently demonstrate his intentions to humanity. One could call it a theology that promotes the reasonableness of the miraculous. Christianity thus functions to guide people in a Newtonian, Lockian universe, discernible through the senses and intelligible to the mind that approaches it empirically. Natural religion sets the stage, and revealed religion becomes the star performer.
If I may offer a minor quibble with this analysis: For the theistic rationalists, reason was the star performer. Revelation was designed to complement or support reason not the other way around. Though these "rationalists," while they all gave reason its due accord, probably differed on the exact proper relationship between reason and revelation.
The site reproduces some of Wright's work:
“There were, in short, two kinds of rationalism in religion in the eighteenth century. One was Deism, which maintained that the unassuming intellectual powers of man can discover the essential doctrines of religion: the existence of God, the obligations resting on men of piety towards their Creator and of benevolence towards one another, and a future state of rewards and punishments. For the true deist, these tenets of Natural Religion were enough, without any doctrines of Revealed Religion. The other kind of rationalist agreed with the deist that there is such a thing as Natural Religion, but denied its adequacy, insisting that it must be supplemented with additional doctrines which come to us by a special divine revelation of God’s will. We shall never understand the religion of the Age of Reason until we recognize that, from the point of view of that century, the difference between these two kinds of rationalism was simply tremendous. We have been led to suppose that because both groups believed in Natural Religion, they were, after all, pretty much alike. It is historically much more nearly correct to say that because one group accepted the Christian revelation, while the other did not, the gulf between them was considered to be unbridgeable.”
“[From the point of view of our supernatural rationalist forebears] Revealed Religion is as rational as Natural Religion, not in the sense that its principles are discovered by the bare use of reason, but in the sense that reason accepts them and approves them as soon as they are known.”
“Here, then, are the essential principles of what we have called–for lack of a better name–‘Supernatural Rationalism.’ Like the deists, the supernatural rationalists asserted the validity of Natural Religion, arguing for the existence of God largely in terms of a Creator who set the heavenly bodies moving harmoniously in their orbits. Unlike the deists, they also asserted the validity of Revealed Religion, which may present doctrines that are above reason, but not contrary to it. Like the deists, they assumed that acceptance of the claims of a particular religion to be a divine revelation is solely a matter of historical evidence and logical analysis. Unlike the deists–and skeptics like Hume–they were persuaded by the historical evidence for Christianity, especially the miracles. Other bases for Christian faith were set aside; its claims do not rest on religious experience, or on tradition, or on the authority of the Church, or on the witness of the Spirit, which had once assured the Puritan that the Bible was truly the Word of God.”
Again I would note the reliance on reason led some of these "supernatural rationalists" to conclude that only parts of the Bible were "rational" and thus "divinely inspired" by a benevolent, rational God. And moreover, reason led many of them to deny the tenets of Christianity's historic orthodoxy, i.e. original sin, the trinity, incarnation, atonement, etc.
On miracles, again reason was the test. Only miracles recorded in the Bible that had some kind of rational basis were accepted as valid. Some unitarians like Priestley and Adams accepted the Resurrection, not of an incarnate God who atoned for man's sins ascending to Heaven, but the act of a benevolent God doing for the most moral man (Jesus of Nazareth), what He may one day do for all good men, perhaps for all men.
Ben Franklin, who, like many of these rationalists, denied the trinity, elevated reason over revelation and held certain things in the Bible were impossible to have been given by divine inspiration, nonetheless believed in bodily resurrection. From a self-epitaph he wrote:
The Body of
Like the Cover of an old Book,
Its Contents torn out,
And stript of its Lettering and Gilding,
Lies here, Food for Worms.
But the Work shall not be whlly lost:
For it will, as he believ'd, appear once more,
In a new & more perfect Edition,
Corrected and Amended
By the Author.
He was born on January 6, 1706.
Franklin also believed in the turning of water into wine at Cana as a "rational" miracle.
It is a mistake, some religious conservatives make, to then note "look at how religious even the Deists Franklin and Jefferson were," and therefore conclude all of the other Founders were traditional Christians. No, Jefferson, Franklin and John Adams all clearly explicated their religious beliefs and all were agreed on these basics. And though Washington, Madison, Wilson, G. Morris and Hamilton (before the end of his life) said and did things that contradicted Thomas Paine style "strict Deism," their words and deeds were entirely consistent with this more moderate hybrid religious system (what Conrad Wright terms "supernatural rationalism") about which Jefferson, Franklin and Adams commonly wrote.