I often wonder why the civil war is so called. In the usual meaning a civil war is war fought between two, occasionally more, factions for the control of a nation.
But in 18612 the Confederate states were not fighting to take over the entire Union. They just wanted to become an independent nation. So it was a war of independence or a war of secession.
In the British Empire we didn’t call the Boer war the South African Civil War. Nor The third Anglo-Afghan war the Afghan civil war. Nor the Indian Rebellion the Indian Civil war. They were wars of independence - parts of the Empire wanting to leave. Just as the American so called civil war was part of the Union wanting to leave
It is only not a civil war if you accept that the Confederacy was ever a separate state. It does not matter what the losers said they were fighting for; officially it was a rebellion.
To elaborate, a civil war (non-international armed conflict) need not be to take control of an entire region. Trying to break away still qualifies. A war to break away from an empire could still be of an international nature, though, like the American Revolution or the Algerian War or the Indian Rebellion. Nobody says the Algerian Revolution was a French civil war, especially since they won.
Depending on your perspective, the Civil War was the War of the Rebellion, The Great Rebellion, the War for the Union, the War Between The States, the War for Southern Independence, the War of Northern Aggression, or the Late Unpleasantness.
At the time of the war and shortly thereafter, northern writers and historians tended to use the War of the Rebellion or The Great Rebellion, where southern historians tended to use the War of Southern Independence. The War of Northern Aggression became poplar in the south during the 20th century.
Abolitionists called it the Slaveholder’s Rebellion.
As per the OP, the Confederacy went out of their way to avoid using the term “Civil War” and instead referred to it as “War between the Confederate States of America and the United States of America” in official documents. The War of Separation or the War of Secession were also used by the Confederacy.
The winner gets to write the history books, and the North won, so they mandated the term Civil War to emphasize that it was an illegal war (from their perspective) and not a war of independence. If you prefer to call it the War of Yankee Aggression instead, there are some folks in the South who even to this day will still agree with you.
Officially, there is no one single name of the war in government documents. The War of the Rebellion appears in a lot of government documents in the late 1800s and early 1900s. The Civil War and The War Between The States are both in use throughout the 1900s and both are still used today.
I’ll suggest that had the Confederacy been successful in seceding and today was an independent nation viewed internationally as co-equal with e.g., the USA, Mexico, or Canada, both the rump USA and the CSA would have different names other than “Civil War” for the conflict.
As kinda noted by others above, “Civil War” is a circumlocution carefully designed to minimize the shame and blame. It treats both carefully unnamed parties as morally and politically equal actors. Call it a (mostly failed?) effort at reconciliation.
I think the fact that Algeria was a French possession, and not classically part of France makes the war a rebellion, but maybe not a civil war. Now if Brittany had tried to break away, that would be a civil war.