I think the OP meant the literal definition of "Civil War", which is multiple sides within a country fighting for control of the government, which it wasn't. The Confederacy wanted to leave and govern itself, making it more closely related to The Revolutionary War (a "rebellion").
If you were in Maryland or Kentucky or Missouri or some other places, the American Civil War was very much a civil war.
South Carolinians thought they were making a revolution in 1861, but when being revolutionary came to be seen as a bad thing, they started pushing "The War Between the States" with some holding out for "The War of Northern Aggression."
For the US government it was originally "The War of the Rebellion," but "The Civil War" came to be regarded as more neutral.
For me, there were enough brothers fighting brothers and families torn apart and enough real aggressiveness on both sides to justify calling it a civil war.