Generally when considering Civil Wars it comes to me as indicative of a war by opposing sides with different positions for how the state conducts its affairs.
In the case of The American Civil War it seems that its focus was on whether the Southern States would or would not split from the Union.
Wouldn't it be more appropriate to name the conflict differently?
There's a lot of debate on 'what' the war was, which has carried on to today. And revisionism is a hotbed topic between the majority of America and regional nationalists like the League of the South, or the Sons of Confederate Veterans. (Let alone the overtly racist KKK.) There are more than a few places in the rural south where mentioning "The Civil War" will get you into an argument over the 'true' name, "The War of Northern Aggression" or "War between the States."
Now putting the context aside...
The answer to your question obviously depends on exactly how 'Civil War' is defined. Merriam Webster defines it as, "A war between groups of people in the same country." As does Oxford Dictionaries.
More importantly, we can look at the definition in terms of international law...
"The common scholarly definition has two main criteria. The first says that the warring groups must be from the same country and fighting for control of the political center, control over a separatist state or to force a major change in policy. The second says that at least 1,000 people must have been killed in total, with at least 100 from each side." ^2
(Edward Wong, A Matter of Definition: What Makes a Civil War? And who Declares it so?)
Finally, we can see that it fits the American legal definition...
Civil war exists when two or more opposing parties within a country resort to arms to settle a conflict or when a substantial portion of the population takes up arms against the legitimate government of a country. Within International Law distinctions are drawn between minor conflicts like riots, where order is restored promptly, and full-scale insurrections finding opposing parties in political as well as military control over different areas. When an internal conflict reaches sufficient proportions that the interests of other countries are affected, outside states may recognize a state of insurgency. A recognition of insurgency, whether formal or de facto, indicates that the recognizing state regards the insurgents as proper contestants for legitimate power. Although the precise status of insurgents under international law is not well-defined, recognized insurgents traditionally gain the protection afforded soldiers under international rules of law pertaining to war. A state may also decide to recognize the contending group as a belligerent, a status that invokes more well-defined rights and responsibilities. Once recognized as a belligerent party, that party obtains the rights of a belligerent party in a public war, or war between opposing states. The belligerents stand on a par with the parent state in the conduct and settlement of the conflict. In addition, states recognizing the insurgents as belligerents must assume the duties of neutrality toward the conflict.^3
(West's Encyclopedia of American Law)
Summary: It's very clear that though the Confederacy's war aim was not to seize control of the entire government, it's still accurate to refer to separatist movements as 'Civil Wars.'
Sources:
1 - I use to be a member of Louisiana's Sons of Confederate Veterans and a serial 'Lost Causer' in my misguided teens. I live in Mississippi now, and I've been in countless arguments with Southern friends over this very topic.
2 - Edward Wong, A Matter of Definition: What Makes a Civil War? And who Declares it so?
3 - West's Encyclopedia of American Law, second edition
EDIT: Added Summary