As I said in a comment cousin to this one, the reasons that Lincoln entered the war aren't equivalent to "the reason the war happened". What you call the "cause" of the war depends on what level you're looking at.
The direct cause of the war was the Fort Sumter altercation.
The reason the Fort Sumter event occurred was that both sides were angling hard for a war they thought they could easily win.
The reason the Union wanted to fight the war was (mostly) that its leaders had strong political inclinations and economic incentives to reject the right of the Southern states to secede, and to use military force if necessary to maintain the union.
The reason the South seceded was primarily that they believed (mostly wrongly) that the North wanted to end slave ownership, which they viewed as both economically destructive and a violation of their rights.
While it's technically true that the story is more complicated than just "the war was fought over slavery", that doesn't change the fact that the existence of slavery was the most significant point of strife in the run-up to the war. But for slavery, the war would not have happened. There are few, possibly no other institutions in the America of the 1850s you can say that about. So it's absolutely correct to say that slavery was the reason the Confederates fought the war, even if it's not the reason Lincoln did. I think that gets you most of what people want to say when they say the war was over slavery.