Enhanced Edition, GOG version. When I press register game from launcher, it wants me to login to my account on this site. I login to here with my gog credentials, so I tried that in the launcher but it gives check your connection message. I suspect the server for this is down but maybe it doesn't accept the gog credentials and I need to make a new account on this site.
Ah. I think I see the issue. As far as I know, registration is no longer required, since the page no longer even exists. (I believe that tab on the launcher is a relic from a much earlier time). You should be able to simply play the game.