Americans generally refer to their 1861-1865 war as The Civil War. More technically it is the American Civil War (ACW) to differentiate from all the other civil wars. Citizens of the USA call themselves “Americans” although really the term could be applied to anyone from North or South America so perhaps it should really be the United States Civil War (but nobody calls it that). The horribly generic “War of Rebellion” was used for the official title of the published records of both sides, more commonly referred to as the OR (Official Records – shorthand for the wordy The War of the Rebellion: a Compilation of the Official Records of the Union and Confederate Armies). Some folks, particularly in the South, prefer other names so they can avoid calling it a civil war. War Between The States is the most common term and was championed (albiet not until after the war) by many ex-Confederates. The more embittered types call it the War of Northern Aggression, Second American Revolution, or War for Southern Independence. More accurate and unbiased terms like the Union-Confederate War, War of the Southern Rebellion, War of the Confederate Rebellion, or (my personal favorite) War of the Southern Secession don’t seem to have much traction.
Although I still usually call it the Civil War because that is the most common name for it, I have come to dislike the term. The Roundheads and the Cavaliers fought an English Civil War. The Catholics and Protestants fought an Irish Civil War. The Spanish had a civil war in the 1930s, the Chinese in the 1930s and 1940s, and the Lebanese in the 1980s. What do all of those wars have in common? Whichever side won would control the entire country. That was not the case in the American Civil War; if the CSA had won they would have achieved independence from the USA. Rebellion, revolution, war of secession, or war of independence would all be far more correct.
Of course, the other side is fuzzy too. The American Revolution (of 1776 fame) included an aspect of civil war, with Tory/Loyalists who stood by England during what was otherwise clearly a war for independence. The French Revolution was actually a civil war between several factions which can be oversimplified as pro-monarchy and anti-monarchy. The Russian Revolution was very much a civil war between reds (communists) and whites (monarchists and democrats). The term “revolution” itself is a bit murky since it can refer to a something that may or may not be sudden. Merriam-Webster offers both “a sudden, radical, or complete change” and “a fundamental change in political organization, especially the overthrow or renunciation of one government or ruler and the substitution of another by the governed.” Perhaps it is best to say that revolutions often trigger civil wars.
Merriam-Webster broadly defines a civil war as “a war between opposing groups of citizens of the same country” which would encompass both struggles for total control of a country and attempted independence by part of a country. I do not like that both types of war are lumped together under a single term like that, but going by the dictionary the common usage of American Civil War is correct.
The war was commonly called a civil war before and during the years it was fought. Lincoln repeatedly used called it a civil war, perhaps most famously in the Gettysburg Address. The term at the time was not solely used by the North. After Lincoln’s election, the Vicksburg Daily Whig referred to “disunion, with all its attendant horrors of rapine, murder, and Civil War…”