But the Jedi vs. Jedi was a major factor in the war.
Put it this way, you've heard of the Clone Wars, correct? Everyone refers to them as the Clone Wars, correct? So let me ask you this: Was it only CLONES that fought in those wars? Was it only CLONES that killed, or were killed during those wars? Was it CLONES who were the leaders of both the Separatist side and the Republic? Was it a CLONE who masterminded the entire thing?
Just because a war is referred to be a certain name doesn't mean that everyone and everything associated with it was the reason for the name of the war.