No one "shot first". Saying "Han Shot First" implies that Greedo got a shot off. He didn't. Han shot. End of discussion.

You saw Han move his head, though, right?

In the original release, Han said the line "I bet you have," and shot Greedo from under the table. Greedo didn't even draw his weapon. In later releases, George Lucas edited Han's head to jerk violently to the right "dodging" Greedo's shot and killing him in "self defense". Bullshit. Han Solo shot Greedo, showed no remorse, and had one of the most badass character introductions ever.