Actually I'm still formulating an opinion which is why exercises like this are so useful. Should a war be judged by why the people who started it did so or by what the people fighting were fighting for? For example, was the reason for the American Civil War an extension of the battle between Jeffersonian and Hamilton politics, defense against an invader or a fight to end slavery? The answer will be different depending who you ask, is only one correct or could all be possibly correct?
I would classify wars in terms of the casus belli, as perceived by the opposing sides. The people who actually went to war were motivated by religious issues during the Crusades. That they were being cynically manipulated into the war by people with ulterior motives was beside the point. People died for their religion, not just the person who was the King of France or the Pope. The ultimate goal of the war was to put Christians in charge of the Holy Land.
In the case of the Civil War, you can look at all sorts of motives, but the one clear motive that was used to rally the troops was the issue of slavery. That is stated clearly in the
Declaration of Causes of Secession and had been the subject of intense public debate during the election of Lincoln, which was the event that triggered the secessions.
Now, just to show I'm not entirely dodging the question, I would define a religious war as one where the decision makers went to war for religious reasons. But I'm not sure that my definition is correct.
I think that 9-10ths' point about multiple causation is a good one, but I would not classify wars as non-religious or religious on those grounds. I would look at the issues that people thought they were fighting the war over, not the realpolitik machinations behind the scenes. A war is religious if the people who fight in it would not be fighting but for some religious issue that they think is at stake.