You specify settings the game must be played on but there is no way to tell if anyone is actually using those setting. Either all runs must show that those settings are on before starting or you should make any settings allowed.

I noticed the runs on the board are actually all new game + runs. I was practicing offline with Twitch and Tremor but it seems everyone is using Admonisher because of the settings.

It took me a while to look it up, but is it correct the only way to unlock Admonisher is by clearing the game 14 times?

Yes, you will have to beat the game 14 times. I've not found any other way yet.

And I now realize that the rules are indeed not very concrete or shown well. (thanks for the heads up everyone.) I will clean this up in the days to come.