Let's look at your original question: "If the South was fighting for slavery during Lincoln's War, who was fighting against slavery?" You're right that the South was fighting for slavery. The South seceded to defend slavery. The South started the war. So the reason for the war can easily be connected to slavery. But you also asked if the North was fighting to end slavery. They were not, not directly.
Let's just stipulate that you are correct so that we can once and for all dispose of the notion that Lincoln's War was fought for some high moral humanitarian purpose like “freeing the slaves.”
But fight and kill Lincoln and his backers did for a purpose which was, for them, desperately important. And the reason: money.