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And can I only view the notifications within Discord?

As far as I know that is the only way!

I think I accidentally logged in via Discord using my master key. I'm not sure if I really did that or that I used my active key. But I guess I need to reset my master key now, just to be safe. Right?

I don’t understand where you did log in with your masterkey! If you have any doubts, it is probably best the change all the keys!

If I click on a link from Ginabot it opens in Discord. So I had to login in Discord to be able to reply. That's where I used a key to login.

Mmh, it opens it on busy.org here.