I (20f) found out my boyfriend (21m) was flirting with another girl.

My boyfriend and I have been together for a year and a half. We are in a long-distance relationship.
We are quite comfortable together, obviously there are periods of ups and downs, but nothing unmanageable.
During the Christmas holidays, however, I discovered something that destabilized me a lot, and I don’t know what to do.

I will call the girl in question Mary, for reasons of privacy.
She and my boyfriend met around the same time I met him.
The situation worsened last summer: my boyfriend and I faced a difficult period, we often discussed for various reasons.
We decided to visit his parents in his hometown, where this girl, who had also returned to the city to visit her relatives, was also staying. They both live in different cities for college.
I already noticed strange behaviors before I joined him at his parents’ house, mainly because he went out almost every day with her, and once he even returned home at 4 am.
When I joined him at his parents house , Mary disappeared for quite a few days, even though I saw that every now and then they chatted and when he did, he answered her when I was not next to him (for example when I went to the bathroom or was in another room).
On the night of our anniversary, instead of spending time only with me, he wanted to go out with me and her to give her a present (a traditional italian sweet) .
That night I confronted him and explained that the situation was degenerating and he could no longer behave like that.
He tried to justify himself by saying that he did not do it with malice and that he even wanted to present her to me.
That time I decided to accept his apology and move on.

Months went by and I still had a bad feeling regarding this situation.
During the Christmas holidays I joined him in the city where he studies to celebrate New Year’s Eve together.
I know it’s not nice what I did, but to confirm that mine were only doubts and not the truth, while he was sleeping I checked the chats on his phone: I found out that both of them were explicitly flirting with each other, that he was sending her almost the same photos that he was sending to me, complimenting her and often he was the one to first approach her in a conversation. He even once told her that he had dreamed that both were on the bed kissing each other. In addition, last summer he had confided to Mary that he wondered how things would have gone if he had gotten engaged to her and not to me. I want to point out that their flirting continued even months after what happened in the summer.
Beyond that I found no explicit declarations of love or things that went beyond flirting.

That same night, at the mercy of tears and especially of anger, I had a conversation about this with him.
I don’t write everything here because the post would get even longer, but in the end he justified himself by saying that he didn’t realize that what he was doing was wrong and that he did everything just to be a good friend to her, and had no intentions other than friendship. He admitted that he had a moment of weakness that summer because things weren’t going well between us, which is why he wrote those things to her.
He even started crying.
I told him that I would forgive him if he ended his relationship and all contacts with Mary, and that he no longer had to allow himself to behave in this way, because next time I would have dumped him without any explanation.
We have moved on and for now he is keeping his promises, but I often wonder if I was right to forgive him, and if what he has done can be considered in all respects a betrayal.