I think when the thought no longer bothers me, I have forgiven. When I could look at a situation that hurt me so much, and think look how it helped me grow. When someone mentions that person’s name, and instead of thinking of how much they hurt me, I wonder how much hurt they have in their life to act like that. I recently read an article by Rabbi Aryeh Kaplan. From what I understood, he was saying how everything in our lives are meant to happen. The good, and the bad. Everything is part of Hashem’s plan to shape your life so you can fulfill your mission, and become a better person. So when someone does something against you, they are only a messenger. It’s sad they chose to be, but their judgement is in Hashem’s hand. Learn from the bad, but don’t waste your anger on it.
Also, I think the anger at how the friend won’t apologize, is just an added anger at your friend. you sound like you’re still quite upset at the whole incident. If you weren’t, why make such a big deal over whether your friends ask for forgiveness?
For the friend who cut off ties, I had that. You can try the 3 times to apologize (Sincerely. Even if it seems so irrational, just apologize for whatever you may have done, and explain that your friendship means so much, that such an ending hurt. Be prepared to let your friend go, but ask for forgiveness. )