Han solo didn’t “shoot first”, he was the only one who shot, period.

Greedo didn’t shoot the wall (and miss a 6ft target sitting 2ft away), that was added (poorly) many years later for a special edition.

They changed the movie because in the interim years, George Lucas was feeling like it made Han Solo look less “noble” and more scoundrel-like.

Greedo was pointing a gun at him and talking about claiming his bounty. How exactly was he going to go about that without shooting Han?

So yes, Han shot first, as in, shot before Greedo could, which he was certainly intending to do.