I dislike the term “civil war.” The South did not want to take over the government (example: English civil war), they wanted only to withdraw from the Union. I’d prefer the term “Second American Revolution.”
Maybe we call them both the "civil wars" rather than "revolutions" because the revolutionaries didn't succeed. There's a resistance to labeling failed uprisings as revolutions, a term which is apparently reserved for more successful rebellions.
And "Second American Revolution" still implies one political entity. Maybe you'd want to call it the "Confederate Revolution" or the "Southern Revolution" or the "Slaveowners' Revolution"
Also, I'm not so sure the line between struggles for control over one government and struggles for division into two governments is all that straightforward. If the two camps in the English or Russian Civil Wars couldn't defeat each other and had to settle for half a country or if Jefferson Davis had been able to achieve independence by deposing Lincoln the line would be pretty blurry.