Texas river cooter (Pseudemys texana)
Another river cooter, with a leaf on its shell, and two red-eared sliders (Trachemys scripta elegans)
The word cooter is ultimately African. "Kuta" means "turtle" in Bambara (now spoken mainly in Mali) and in Maninka (a major language of Guinea), and possibly in other Mandé languages.
Through a folk belief that turtles copulate for two weeks at a time, a folk etymology of cooter arose in which the back-formation verb "to coot" referred to the supposed fortnight of sexual congress, and the name of the turtle came to be thought derivative from their alleged behavior. The OED documents this usage from the middle of the 17th century. No one knows why these turtles were thought to have such prodigious tenacity in coupling. Maybe it just came from the idea that turtles were slow about everything.
But they are not slow in getting off logs when you approach. In fact, the name of their relatives close, the sliders, refers to the slider habit of moving, and moving quickly, into the water if approached. Cooters do the same.
The word slider is not quite right itself. It's more like plopping than sliding. Oftentimes I become aware of an unseen turtle by the sound of its having fallen into the water with a sound like a brick had somehow pushed itself off a log. Not a sliding noise, exactly.
There has been some attempt to regularize the terms cooter and slider in field guides and such, but in common usage both words refer--depending on where you are--to any of a wide variety of aquatic turtles. When I was growing up in Texas cooter was a word for any river turtle. (It was also an occasional human nickname. I don't know why. I knew a kid named Cooter.) I don't recall ever hearing the term slider, though I am sure it was also common in some places. Nor was I aware of any sexual innuendo involving cooters, though I have read that in recent years a cooter (turtle) festival in some little town in the south had to be cancelled because some members of the community believed the word was offensive. Though it was hard for me to figure out why from the newspaper article, I think the people who objected to it considered it a substandard word for vulva.
You can't predict the evolution of words.