Why did George Lucas change it so Greedo shot first?

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.”

It is worth noting that George Lucas claims that Han never shot first:

The controversy over who shot first, Greedo or Han Solo, in Episode IV, what I did was try to clean up the confusion, but obviously it upset people because they wanted Solo [who seemed to be the one who shot first in the original] to be a cold-blooded killer, but he actually isn’t. It had been done in all close-ups and it was confusing about who did what to whom. I put a little wider shot in there that made it clear that Greedo is the one who shot first, but everyone wanted to think that Han shot first, because they wanted to think that he actually just gunned him down.

Combined with the more recent quote in the Washington post, it is fair to say that Lucas edited the movie in order to make it clearer that Han Solo would "let them have the first shot".

You're going to find that many of the truths we cling to depend greatly on our own point of view.

While Lucas has a reputation as somewhat of an unreliable narrator when speaking of his early Star Wars plans, I don't think there is enough evidence to say definitively either way who shot first in the 1977 film.

The two best sources of information are the original film, and the original scripts.

In the original film, no blaster bolt appears at all, but there are 2 flashes of light.

First, there is a flash of light over Han Solo:

Then the shot reverses, and we see Greedo, followed by a screen flash and a pyrotechnic.

You can check it out yourself on Youtube.

Neither interpretation really lines up with blasters in the rest of the film: Han is firing his gun in close-ups only a few scenes later (while they take off from the spaceport), and there is no full screen washout effect. And most (but not all) blaster shots leave some damage to the walls they hit.

Scriptwise, Peter Mayhew's original shooting script says this:

Suddenly the slimy alien disappears in a blinding flash of light. Han pulls his smoking gun from beneath the table as the other patrons look on in bemused amazement.

Some people have pointed to this script as "proving" that Han shot first (including Mayhew in that tweet), but there's no smoking gun: all it says is that there was a flash of light, and that Han "won" the shootout. This mostly lines up with the film we have access to, but says nothing as to whether Greedo fired. (Note also that Greedo is called "Allen" in the script)