First Barton lies about [John] Adams’ faith, positing that he was a ‘Trinitarian Unitarian.’ Unitarianism, however, is based on the rejection of the Trinity and noted Unitarian thinker James Freeman Clarke writes that “Unitarians, strictly speaking, are those Christians who reject the Church doctrine of the Trinity, and do not believe that Jesus is God the Son, equal with the Father, or that he is the Supreme Being.”

Actually, it’s not obvious to me that Barton was lying (or confuse, or sloppy) when he said that John Adams was “a Unitarian who believed in the Trinity”.
Those familiar with Christianity (and with the history of Christian “heresies”) can appreciate that it’s possible to believe in the Father, the Son, and the Holy Ghost while simultaneously rejecting the Nicene formulation of the Trinity that describes the inter-relationship and the specific characteristics of “Big Daddy, Junior, and Casper.” Or in other words, you can reject the Trinity™ but still believe in some sort of Celestial Troika. To the best of my knowledge, for example, Mormonism emphatically repudiates Trinitarianism, and yet speaks of the Father, Jesus, and the Spirit.
But at the colloquial level, it’s normal to conflate “believing in the Father, Son, and Holy Ghost” with “believing in the Trinity.” And since Jon Stewart is secular, from a non-Christian background, and offering “edutainment” rather than meticulous scholarship, it seems pretty defensible for Barton to use a loose, colloquial definition of what the Trinity means.
For that matter, Jon Stewart (and RightWingWatch) seem to be using a modern definition of Unitarianism — God is One, Jesus was not divine at all, and “Holy Spirit” is simply a synonym for “God”.
But from Googling and wikipedia, I get the impression that some 18th-century Unitarians (though not necessarily John Adams himself) would’ve accepted that Jesus was in some sense an Incarnation of the Divine, and was bodily resurrected, while totally rejecting the doctrine that JC was coeternal with the Father.

The Foundations class is unlike anything offered at secular law schools, its purpose being to guide students toward a “Christian worldview” of the law. In the 2008-09 academic year, the required texts included David Barton’s Original Intent, which Barton’s website describes as “essential resource for anyone interested in our nation’s religious heritage and the Founders’ intended role for the American judicial system,” and Francis Schaeffer’s Christian Manifesto.

They also seem to be instructing law students that they should have encouraged Lisa Miller to go to Nicaragua. They even put it on an exam.

The Foundations class is unlike anything offered at secular law schools, its purpose being to guide students toward a “Christian worldview” of the law. In the 2008-09 academic year, the required texts included David Barton’s Original Intent, which Barton’s website describes as “essential resource for anyone interested in our nation’s religious heritage and the Founders’ intended role for the American judicial system,” and Francis Schaeffer’s Christian Manifesto.

They also seem to be instructing law students that they should have encouraged Lisa Miller to go to Nicaragua. They even put it on an exam.

If I had to take a guess at what Barton is talking about, it may be that one particular Unitarian minister published a theological polemic against the Holy Ghost in 1839

When I wrote this, I hadn’t yet read mlargles’s blog post, which suggests that a 1839 speech by Unitarian theologian Andrews Norton, entitled “The Latest Form of Infidelity,” is the thing that Barton vaguely referred to in his Daily Show appearance.
I haven’t fully digested Norton’s essay yet, but as far as I can tell from a quick skim, he was one of those Unitarians who in some sense “believed in the Father, the Son, and the Holy Ghost”, yet rejected the Trinity.
I think it may be the case that Norton regarded the Father, the Son, and the Holy Ghost as being (in modern Internet slang) “sock puppet” identities of the One God — or perhaps more like Peter Sellers playing the U.S. President, the British officer, and the German scientist in Dr. Strangelove.
The idea that Jesus the Son of God was a “sock puppet” or “role” performed by the Supreme Being is obviously incompatible with any traditional formulation of Trinitarianism, but is also at odds with the “secularized liberal Christian” notion that Jesus was merely a gifted rabbi who attracted a lot of followers, but not Divine in any special sense.