After reading this question:
Is there a reason for this?
I hate to answer the question with a wikipedia quote, but here goes (from the page you linked):
Thus, the phrase "Han shot first" is a retort to director George Lucas' explicit cinematographic assertion that "Greedo shot first."
That's it, simple as that. It's the opposite of "Greedo shot first" rather than an actual description of what happened in the original release.
In the Special Edition, George Lucas changed the cantina scene so Greedo fired but missed wildly, and then a split second later Han's shot (in the original footage) hit Greedo. This crippled the established character of Han, who was originally written as a scoundrel with a strong sense of self-preservation.
So "Han shot first" is how we complain in response. It's easier/quicker than saying "Han solo shot Greedo at the first opportunity he could to save himself because that's what a smuggler running from a crime lord he cheated would do. A professional (though inexperienced--see the Tales from the Mos Eisley Cantina book) bounty hunter would not miss while sitting down with a blaster pointed straight at the victim who is sitting still 2 feet away."
A thought occurred to me today that helps explain it.
The alternative to "Han Shot First" is "Greedo Never Fired"
The second is definitely more accurate: Greedo did NOT fire in the original theatrical version. That said, the fan outrage was geared towards inciting a more general outrage or at least making sure everyone knew WHY the fans were outraged.
Greedo is a minor character who spends less than 5 minutes on screen, is hardly referenced in the EU, had no memorable lines, and is only notable for being a complete idiot.
Han was played by Harrison Ford. He was in all 3 movies. He has an extensive presence in the EU. His list of quotable lines is longer than a child's arm. He was on virtually every poster. Han is almost as universally recognized as being tied to Star Wars as the lightsaber.
Therefore, their mantra had to feature Han.
Secondly, "Han Shot First" is an action statement - it says something about the character. It is a definite action that he took, and it is clear enough to reference the scene implicitly. "Greedo Never Fired" rather says what a character didn't do. It doesn't give us a clear picture of Greeo's character, it doesn't say what he would have done, it simply says he didn't fire.
The choice between an action statement and a passive statement is clear. People like action, not inaction. "Han Shot First" is simply a stronger statement, and it is much more memorable.
I think there are two different ways to parse the phrase "Han shot first":
Among the multiple beings who fired their guns, Han was the first.
There were multiple beings who wanted to fire their guns, Han was the first person to do so.
You're reading it as the first way, but if you read it the second way it makes sense. Han and Greedo both wanted to shoot each other, but Han got there first. You could say this in many other cases; for instance, I was going to go to the breakroom for some free donuts, but everyone else got there first. That definitely does not imply that I still got a donut.
Also, note that there's a potential double meaning here. "Han shot first" could be a reference to the 'first' version of the film, in which Han was the only person who shot. In other words "at first, Han shot alone, but later both Greedo and Han shot".
Without getting over analytical, when the original Star Wars showed the confrontation between Greedo and Han, While talking about taking his money the conversation went:
Han: Over My Dead Body
Greedo(while pointing a gun at him): That's the Idea. I've been looking forward to this for a long time.
Han(while secretly unsheathing his weapon under the table): I'll bet you have
And Han shoots him under the table.
Its a cool scene and gives Han credibility in a dog eat dog situation.
If you change it and use crappy CGI to make Greedo shoot first, Han dodge and return fire, it is self defense. It portrays him differently and makes the scene SO much less cool.