I have been having the exact same issue, I have wasted over 10 hours alone trying to both find fixes for this, along with trying several browsers, the same ones you listed. I have tried searching google for so many hours, found only one suggested fix, and it didn't work, that button for logging in is broken. Also my name is not listed at the bottom of the quest page screen, something is wrong with the page, not detecting you as logged in, even though you supposedly are.
I hope they fix it, because I've retried and finished that mini-game over six times, only for the login button to never work. (I thought I was the only one who was confused as to why it seems to work for everyone else and not me.)
Tried IE, inquisitor quest clicking didn't work right.
Tried Firefox, finished all successfully, login button doesn't work.
Tried Chrome, finished it successfully, but login button once again does nothing when clicked.
