One piece of evidence for this is that they typically used non-specific names for God that were minimalist enough that all religions could view them as speaking to their own particular religion. Thus in the Declaration of Independence you get phrases like Nature's God and Creator and Divine Providence. George Washington, in particular, had a large number of phrases that all worked to the same effect, as a sort of lowest common denominator deity, including the great governor of the universe, the supreme disposer of all events, and the Almighty ruler of the universe. Thomas Jefferson likewise spoke of the Great Governor of the world.

Another piece of evidence for this is that these men often used the preferred terminology for God of the people they were addressing. This was true even of religions that were not of the Abrahamic variety, religions which were very different like the beliefs of the American Indians at the time. When speaking to the Indians, these men routinely spoke of the "Great Spirit", which they viewed as merely another title for the one God that they all believed in. In an 1803 letter to the Choctaw Indians, for example, Thomas Jefferson closed his letter thusly:

But we thank the Great Spirit who took care of you on the ocean, and brought you safe and in good health to the seat of our great Council; and we hope His care will accompany and protect you, on your journey and return home; and that He will preserve and prosper your nation in all its just pursuits.

LIkewise in an 1806 letter to the Cherokee Nation, he spoke even of praying to the Great Spirit:

My children, I thank you for your visit and pray to the Great Spirit who made us all and planted us all in this land to live together like brothers that He will conduct you safely to your homes, and grant you to find your families and your friends in good health.

George Washington had already set this tone in his administration a decade earlier, closing a letter to the Cherokee Nation in 1796 by saying, "I now send my best wishes to the Cherokees, and pray the Great spirit to preserve them." Washington was a Master Mason, of course, and one of the tenets of Freemasonry is that all religions are more or less the same, all valid ways to God.

James Madison used the same convention and went so far as to say identify the Great Spirit as the creator of us all in an 1812 letter:

The Great Spirit has given you, like your white brethren, good heads to contrive, and strong arms, and active bodies. Use them like your white brethren of the eighteen fires, and like them, your little sparks will grow into great fires. You will be well fed, dwell in good houses, and enjoy the happiness for which you, like them, were created.

It seems clear that, for these leading founders, their theistic rationalism demanded a universal deity that was called by many different names. They all rejected claims of revelation as well, which I think speaks to the fact that they believed in a "lowest common denominator" deity, one that all religions could agree on in its few basic characteristics. I suspect this is what Adams had in mind when he spoke of "primitive Christianity", the very bare core of beliefs that Christianity has in common with other monotheistic religions - the notion of one God who created the universe and everything in it.

All else that we needed to know about God was, to their way of thinking, discernable through the application of reason alone with no need for direct revelation. Thus, we see Jefferson urging his nephew, Peter Carr, to take everything in the Bible and in all other books and subject it to the tests of reason. All that seems contrary to reason and logic is to be discarded, everything that stands up to scrutiny is kept. Reason, not faith, was their ultimate concern.