Can you talk to your dad about it? Feels like you need to talk to him, not your mom, and tell him how it made you feel. I’m sure he didn’t mean to hurt you, but it was a consequence of his actions nonetheless.
As for your mom, she’s from another generation or mindset that thinks it’s ok behavior, or not a big deal. You disagree. Probably best to leave it at that, but it’s fine that your feelings towards fidelity cause you to hurt in this situation.
And it may say something about you too. Remember that it’s their relationship primarily, and your depression is probably unrelated to the affair. It’s a trigger, so try not to take it too personally.
You don’t have to forgive him. Every person is different. It’s that simple.
Went through the same thing when I was around your age- it’s okay to feel mad. Don’t let anyone else tell you how you should feel. At the same time you can’t expect your mom to feel the same way you do.
It was very wrong of your mother to include you in her conversation about your dad’s suspected cheating. You know more than you should about their problems and you’ll never know the whole story. Be angry at your dad for the betrayal and be angry with your mom for dragging you into it. They both need a talking to. Being shitty parents and then telling you everything is going to be okay is wrong. Everything is not okay for you.
I’m going to say something controversial. This isn’t your business. Relationships are really hard. Your mum sounds like she is protecting you but you don’t know the whole story.
You might find what you are looking for by confronting your dad but go into it with an open mind. Be prepared to hear things which won’t fit the narrative you think might have happened.
Ultimately they are 2 adults. People make mistakes and forgive or don’t. Your mum might be giving you one last chance to live as a family. Will it be different? Possibly. Will it last? Who knows.
I’m sorry though. Change is horrible when it involves your family but you are 20 and unfortunately life has a way of rocking your family unit.