Yes, the rumors about this are, that Lucas added Greedo shooting first, to make Han look less cold-blooded. You have answered your question yourself - "Shooting Greedo in self-defense did not make Han look cold-blooded". That is exactly what Lucas did, he made it look like Han killed Greedo in self-defense. But in the original cut, where Greedo didn't shoot first, it was not self-defense. He just straight up murdered a guy who came to ask for money. That was cold-blooded.
Edit: George Lucas said:
“Han Solo was going to marry Leia, and you look back and say, ‘Should he be a cold-blooded killer?’ ” Lucas asks. “Because I was thinking mythologically — should he be a cowboy, should he be John Wayne? And I said, ‘Yeah, he should be John Wayne.’ And when you’re John Wayne, you don’t shoot people [first] — you let them have the first shot. It’s a mythological reality that we hope our society pays attention to.”