In my country/culture the idiom goes "Nero didn't play (or did, people use both versions) the lyre while Rome burned". I don't know what else to tell you. The term being invented or not doesn't matter since the instrument wasn't invented. Your example with Cesar is different. If knives weren't invented by that time (or other stabbing implements) would you still say he was stabbed to death? Language doesn't matter in face of physical reality.
To me, it is the objectively correct version of the idiom, as it involves a historically accurate instrument.