Han shot first came about like you said due to the special edition nonsense. But it was because people remembered the original where he indeed shot his pistol before Greedo could shoot his. So it makes perfect contextual sense. If you watched ANH original copy with no knowledge of the issue, you would come away seeing that Han Solo was held up at gun point and being threatened to be shot but he was able to kill his attacker first by shooting him.
Han shot first before Greedo could shoot him. Seems easily understandable in context.