In the original theatrical release one guy shot first, and in subsequent releases on VHS, DVD and Blue Ray the other guy did.
You may think that Han Solo shot first, but he didn't. The theatrical release of Star Wars was in 1977. 11 years before that Sergio Leone directed, "The Good, The Bad, and The Ugly." The movie begins with three vignettes wherein Clint Eastwood--the good, Lee Van Cleef--the bad, and Eli Wallach--the ugly in turn introduce themselves to the audience.
Lee Van Cleef, playing the character Angel Eyes, makes an unwelcome visit to someone with information and he sits down at the man's table. While he interrogates the man, who is in fear of his life, he eats the man's supper. Eventually, the man goes for his gun, and Angel Eyes shoots through the table killing the man in cold blood. Moments later, he guns down the man's oldest son. The scene makes it clear that this character is one stone-cold murderer. He's Bad.
Fast forward 11 years and George Lucas is making his own film. He's introducing a character and he wants to demonstrate that this character is not a nice guy. Lucas grabs the shoot-through-the-table without warning gag that he and most of his audience has already seen. In moments, Han Solo is seen through the lens of Lee Van Cleef's portrayal of Angel Eyes.
That was then. The kids nowadays haven't seen any Clint Eastwood spaghetti westerns before they see Star Wars. Unlike a generation ago, things are reversed, kids interpret Angel Eyes' gunplay through the lens of Han Solo and Greedo instead.
Lucas now wants a softer, kinder and gentler Han Solo who's more fitting with a lovable rogue, not a stone-cold killer.