Check out this great article at Patrol Magazine by Alissa Wilkinson, a graduate student at NYU, on American political theology that references "theistic rationalism."
Here is a taste:
The secularists—the most dogmatic “separation of church and state” folks—insist the Founders were Deists with little interest in organized religion, working toward a neutral, secular state where religion would have no influence in governance or policy-making. Equally noisy are the “Christian America” proponents, who insist that the Founders were devout Christians with explicit faith in Jesus Christ and established a governmental system based on Biblical principles. Any attempt to extricate governance from these principles is an attempt to destroy the very foundations of the country. References to “God” and “Providence” in the founding documents, such as the Federalist Papers and the Declaration of Independence, are explicit and intentional references to similar evangelical concepts.
What’s confusing is that both camps can support their view with books, films, seminars, scholarly works, magazine articles, and more, all with direct quotations from the Founders themselves. And obviously, both sides can’t be right. So when it comes to the ever-raging debates about the foundations of our nation, which side should Christians take?
In [Gregg Frazer's] doctoral dissertation and some subsequent work, he says—I believe rightly—that these men were neither secularist Deists nor evangelical Christians, but “theistic rationalists,”...
To theistic rationalists, God would not do anything that they would not admire in the behavior of man. Order and morality were the highest virtues. Men had a free will and the ability to be moral, and God ultimately desired all men to live happily.
Religion was important to society in that it promoted morality—and thereby happiness—but the particular religion was relatively unimportant. Because the ultimate goal was a moral society, rather than one in which the “correct” religion was promoted, the Founders created an environment that recognized but did not impose or restrict the role of religion in society. (It’s worth noting that several Christian denominations opposed this idea of freedom of religion, since it would allow many people to practice religions that they did not believe led to the truth.)
...[T]he end result of this emphasis on morality and freedom was that theistic rationalism became the de facto national religion. Most people in early America identified with a Christianity of some stripe, and so these principles also became woven into the fabric of American Christianity and the dominant public desire for morality and order....Only when postmodernism erupted and new voices spoke out in the public sphere—minorities, women, people of other religions or no religion at all—were they challenged, spawning the debate that still rages today.
With this in mind, we can begin to understand the flaws in the views of those on both sides of the debate. Some of the most influential Founders did in fact believe in the value of religion for a moral, organized society—which weakens the position of the secularists. But they also did not believe that a theologically orthodox Christianity was the only or even the best option for promoting that society—undermining those who would have us believe we’re citizens of a “Christian nation.”