Another repost here.
Let me note some folks here disagree with the term "theistic rationalism" to describe the theological system Madison was talking about. So let's clarify: Madison's notes on why he wrote the Memorial and Remonstrance which led to the separation of Church & State in VA, refer to the issue of "what is Christianity?" He noted 1) Arianism and Socinianism as presenting themselves as "Christian"; 2) folks dispute what books properly belong in the biblical canon, and 3) folks practice a form of "Christianity" that doesn't hold the biblical canon as "infallible" but rather that only certain "essential" parts of the Bible are divinely inspired.
Madison didn't believe government had a right to take cognizance of which theological systems qualify as "real Christianity." Back then, as today, the "orthodox" didn't believe a system that denied the Trinity and the infallibility of the biblical canon to be actual "Christianity." Rather they argue it is some "other" religion. Dr. Gregg Frazer has termed it "theistic rationalism." Back then the orthodox theologians said it was no better than "Deism." Those who believed in it, however, termed it "rational Christianity" or "liberal unitarian Christianity."