Boyfriend talks dirty to another girl, should I break up with him?

I have been with my boyfriend for a long time about 3 years now and we are on a long distance relationship but we meet up whenever we can. I was with him the other day and when get off the car to get something, something in me made me check his phone and i went through his chats and saw a conversation of him with another girl with him in college having an inappropriate talk. I got really upset and angry and asked him to drop me back home and that im breaking up with him he started crying and begging me to forgive him and to give him another chance saying that its nothing and that he doesn't have any intentions and he was feeling bad about doing it and wanted to tell me and then when i got angry about him it woke him up and he will stop doing it. I did forgive him and not break up with him but im still hurt and I don't know if im making the right decision by staying with him in a relationship. He said he is going to change and make me trust him again and that he is willing to do anything for me to forgive him.
What do you guys think? did i make the right choice?
should i break up with him? is it considered as cheating?
Because we did have a fight about this particular girl earlier 2 years ago and he told me to trust him that he won't do anything that would hurt me or that i wouldn't like, when i reminded him of what he said 2 years ago he said that he did it just to stop himself from fucking girls or doing anything worse than that and that it was pointless and he is ready to give it up.
When i asked him whats gonna make him stop he said that the feeling he got when i knew and the feeling of loosing me is whats gonna make him stop
but im scared because im a jealous girlfriend and i am scared that when i go back to my country this is gonna be an issue for us because trust is really important in a long distance and i told him that i dont trust him anymore.

what do you guys advice me to do?
Boyfriend talks dirty to another girl, should I break up with him?
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