Thanks nyuszi008, my game is running using that!
I'm guessing that when it's on Steam, that Steam will somehow just run the file without having to type a command in terminal each time.
Interestingly, I transferred both the zip and the extracted zip to the Linux Mint computer, and it seems to be able to run both versions.
However the external level data files that I normally include alongside the .exe on Windows, when I transfer them from windows to linux it doesn't seem like that works here. Perhaps it's related to that permissions thing.