Han definitely shot first in Star Wars!
The whole Star Wars saga has become a bit of a mess when you consider the release of the original film as a stand alone enterprise just called Star Wars, then later rebranded as Episode IV A New Hope in time for the release of The Empire Strikes Back. Then in 1997 the Special Editions came out, with new footage, random CG characters and a whole lot more. And that's before you consider the prequel trilogy.
It's all down to creator George Lucas who just can't seem to leave the thing alone. His constant revisions rarely seem to improve matters, especially when it came to the original trilogy. And one of the most contentious has always been exactly what happens when Han Solo (Harrison Ford) meets Greedo in the Cantina.