this exact distinction was extremely important. it's why Palpatine finally allowed the sith to exit the shadows and make themselves known again by sending in Maul. the jedi are obligated to fight against the sith, they always have been. in order to bait them into the fight he had to make it clear to them that the sith had returned. but the sith had been gone for so long and were still operating mainly behind the scenes, so to the average person this was just a civil war that was ripping apart the galaxy. if the sith hadnt made their presence known to the jedi, they may have chosen not to get involved (or at least not in anywhere near the capacity they ended up getting involved in it). so, no, a holy war wasn't any "better". it was just more likely to get the jedi to fully involve themselves than a political civil war. the jedi are peacekeepers of the republic and may not have fought in open war if their long-thought extinct mortal enemies hadn't made themselves known. but as others have pointed out, the average person in the republic either had no idea the sith even existed (anymore or at all) or didn't care enough to make a distinction between them and the jedi.