In the book, it's not just that Han shoots first.
(Yes, there's a book.)
It's that every single person in the bar except Greedo knew that it was a Bad Idea to let Han get his hands under the table. They even started moving out of the blast zone, smirking.
Han's a tough motherfucker, and Greedo was fucked the moment he signed on for credits. He's a survivor, and the only reason Luke and Obi-Wan lived is because Han is one of the only pilots to enter hyperspace with his shields on. (Costing excess fuel and a minor environmental problem.)
Look at the ambush with Vader at Bespin -- Han doesn't shoot a look at Leia, no one-liners, neither a "you asshole!" to Lando nor a "it's a trap!", he just pulls out his blaster and starts blasting. Every shot is dead center, and it costs Vader something like FOUR force points to survive the attack.