A misunderstanding of the meaning of the word "civil" is what leads to confusion here. Of several definitions, civil means "of, pertaining to, or consisting of citizens." It also means "of the commonwealth or state."

So quite literally, a Civil War is a war of citizens, a war of the state.

Folks seem to mistake civil as simply meaning courteousness or something and that's not the whole story. The only reason that it can mean something like that is because it references society, eg, how one would act in society. Essentially, telling someone to be civil is telling them to not act like a part of the fringe.

Dictionary!