Yes and no. When the southern states seceded from the union, they ceased to exist as States of the American Union. The south had its own president and government through the course of the war. In this sense, there were two countries fighting each other, but what made it a civil war is that the two countries had originally existed as one.
Classification of the conflict has nothing to do with the people fighting, and more to do with the status of the nation(s) they belong to.
Classification of the conflict has nothing to do with the people fighting, and more to do with the status of the nation(s) they belong to.
The United States never recognized that the Southern States could secede and form their own country, and thus, never recognized the Confederate States as a different country. That is sort of why the whole war was fought.
In Skyrim's case, I guess it does depend on who wins. If Ulfric and the Stormcloaks win, it was a war for Independence. If The imperials win, it was a provincial uprising. In either case, it is a civil war, as you have the Jarls, representing the leadership of Skyrim, fighting among each other. The Jarls are basically fighting over whether to remain a part of the Empire, or to break off. The Empire obviously supports one side of the conflict.