Faithful but not in a official relationship

Background: My ex-boyfriend and I were basically college sweetheart. I started messing with him my freshmen year and that lasted until senior year. We were officially together for 1 year and 9 months. I recently just graduated in December but I graduated early which means he is away in college right now. He broke up with me 2 weeks before I graduated bcus he said he wanted to work on becoming a better man for himself and for me. During our relationship, I caught him on 3 different occasions texting and talking to other girls which I forgave him for. And he has became physically abusive on 2 occasions which I also forgave him for. I want to add that I'm not perfect and I'm not the easiest person to deal with either. We all have a faults in relationships.I would say we were in a toxic relationship but his my first real grown up relationship so my decision making wasn't and  isn't the best. 
Currently: We are not official boyfriend and girlfriend but we basically act like it. He doesn't want to put the title but he wants to act like it. The crazy part is that I want to be in a relationship with him again but he doesn't. I've heard you're either in a relationship or you're not. I don't entertain any other dudes at the moment because I'm being faithful to him. His at school 2 hours away and we don't get to see each other often.  So my question is that should I be faithful to him even though we are not technically together? Am I being naive and stupid in love? Please ladies, I need advice but no judgement.