Christian Atheists are people who find the philosophical system presented by Jesus, as they interpret it anyway, one they can follow and recommend others follow, with the exception that they will not call any figure "god," neither Jesus, the (his) Father, nor the Holy Spirit. They believe in Jesus the man only, not any god at all, thus "Atheist." Lately they often criticize Christians for not following the teachings of Jesus.
In the 1960s there was a "somewhat" organized movement in religious and philosophical education (college professors, books, classes) that "somewhat" publicly embraced the term "Christian Atheism." Thomas J.J. Altizer was an associate professor of religion at Emory University. He aligned with the death-of-God movement insofar as saying that it is necessary to recognize the divinity of Jesus was becoming obsolete in modern culture. William Hamilton of Colgate Rochester, Gabriel Vahanian of Syracuse, Paul van Buren an Episcopal minister and associate professor of religion at Temple and several other writers spoke of and aligned more or less with the "movement" described in the April 8, 1966 Time Magazine with the cover, "Is God Dead?"
That was then, this is now. There is less mincing of words. There are more followers of Jesus, or what they think was Jesus, who are more openly atheist. Christian Atheists today often believe other Christians who are not atheist have a learning disability because of the continued belief in a god. They believe "science" has dismissed the possibility of any god. They think that once other people realize there is no god they will follow the teachings of Jesus without further difficulty.
There are many "Christians" who claim to believe that Jesus is in fact "god," but they are otherwise indistinguishable from atheists. They think there is nothing wrong with homosexuality. They do not have a well developed sense of private property. Their ideas on economics seldom use and sometimes violate mathematics. They think they're experts on biology, but are not, and so on.
While Christian Atheists believe in a human (totally human) Jesus, Military Atheists believe in the military and the state. It constantly frustrates them that godly people appear so ill prepared to rule the world. They believe "might is right," and perhaps one or another rendering of the phrase, "power comes from the barrel of a gun," and only threats of violence and death can guarantee the level of cooperation necessary in the world. Especially lately they agree with Christian Atheists that the faithful have a learning disability that prevents them from ordering the world around as needed. Otherwise they sometimes disagree with Christian Atheists on how exactly the world should be ordered. "Regimented" thinking can be especially noticeable in Military Atheists. In the past, science was more in the business of persuading than forcing, but military atheists think their power to enforce is enhanced by their "science."
Again, there are many "Christians" who claim to believe that Jesus is in fact "god." Some of them really only believe in the military though. Of course Military Atheists are not the only people in the military. Some military are members of advanced, civilized religions.
Several schools have long taught to never call anyone "stupid." Avoiding such a term is excellent advice. The term generally serves no good purpose. It only creates needless animosities. Elementary Atheists might well have some degree of learning shortfall though as they prefer not to develop beyond what is actually an elementary school level of understanding of the world around them. Their understanding of biology is from pre-Civil War times. They believe in "science," but they often aren't capable of it. The abstract ideas in advanced religions are beyond their tastes if not their abilities. They misunderstand and misuse argumentative techniques and believe they have "won" arguments. They are atheists because the only concept they have of a "god" is an old man with a long white beard who lives in the clouds. No advanced religion believes in such a god. Some of the faithful do allow that image as an accommodation to children, but that is all.
The Democratic Party has many people who would be considered Christian Atheists, the Republican Party has many people who would be considered Military Atheists. While some of both of those will identify themselves as such, many will not. It would be a mistake to characterize either party by those members. Rather consider their platforms and other published materials.
Part of the recent failing of education is that colleges have large numbers of people who remain Elementary Atheists.