If you want to get it right contextually, in the original theatrical release, Han is the only person that fired. To say he "shot first" would imply that there was another shooter. The saying itself is silly but if people are consistently going to keep saying it and not letting people forget, at least get it right.
Makes perfect sense actually. "Han shot first" works fine. The alternative "Han shot Greedo" would apply to all situations where Han did eventually shoot Greedo, "Han shot first" could imply he shot before Greedo had a chance to.