I think I might’ve found why the ‘New version available, please refresh’ bug (maybe exclusive to Mac users?) happens.
The first time I load, on both browsers, the unity player fails to load and its error message is ‘Failed to initialize content’. From then on, the next reloads of the game all have the ‘New version available’ message pops up. I think what happens is that the update, seen on the first try, fails to ‘initialize’ for some reason, and cannot update, and all subsequent reloads don’t have the correct version of the game.