Maybe a log-out button could be added to the Play Game page?
Perhaps, like on the Character Resetting page, there is a little log-out link. I know the site automatically logs you out of the page if you're idle for too long, but perhaps a way to do it manually?
Just a thought I had. I didn't dig too deep but I did look a few pages back and didn't see this, sorry if it's already been suggested and I missed it. <3