Ultimately, though, the sole motivating force was defense of slavery. Lincoln wanted to keep the union together but that required addressing the thing that split it apart, which was slavery, and ultimately that's something he did. But that's Lincoln.

The CSA's leaders and generals had to justify, at least to themselves, their continued war operations even as the war was destroying their new (fake) country. The only thing they had that motivated them so much was slavery. they were not shy about saying so and they said it consistently.

Trying to get technical over what the war was really about only confuses the matter. The South was determined to do anything it could to preserve slavery, whether forcing it on the Northern states or refusing to acknowledge the validity of the federal government, and all of those things were ultimately going to lead to war.

"States' rights", here, is a nonsense justification for the war (and before the war, for slavery) and not the justification the South gave for engaging in war to begin with just as "secession" is. It was always slavery that they were concerned with, and whether states rights supported slavery in the moment or opposed slavery they always sided with slavery. Here's a sampling of Jefferson Davis in 1860:

Resolved, That the union of these States rests on the equality of rights and privileges among its members, and that it is especially the duty of the Senate, which represents the States in their sovereign capacity, to resist all attempts to discriminate either in relation to person or property, so as, in the Territories—which are the common possession of the United States—to give advantages to the citizens of one State which are not equally secured to those of every other State.

What he's saying, here, is that it's a violation of states' rights (or an establishment unequal rights) to prevent slavery in the territories and in new states. Why does what happens in a new state have anything to do with the rights of (say) Mississippi? Because if enough new free states were admitted to the union they would have enough political power to ban slavery, which is the thing that mattered.