While it does sound to me like a huge line was crossed, I have to say that I think in these matters it's best to have a verbal agreement with your partner. It sounds to me like he became emotionally connected with this woman which, if the agreement the two of you have is monogamy, presents a lot of problems, as you outlined. I'm not surprised that you don't trust him.
What I have found is that everyone is different. For example: in a past relationship my partner was always chatting online and flirting, especially with younger girls. He was always watching porn and just generally being shady, and then refusing to sleep with me. That was an issue. I became jealous and obsessed and monitored everything. I would come back from being away for vacation and all these girls would come up to him when we were out in public (we lived in a relatively small town) and tell him what a 'good time' they had with him while I was gone. With me standing right there. Drove me nuts.
I took that out of the relationship and projected it onto future partners. I assumed that porn and the internet were the enemy and that nobody could be trusted. In relationships things would be great for a few weeks, and then I would realize that they still had attractions to other people and feel betrayed, like I couldn't trust them. Totally insecure. And all the old anger and jealousy would come up.
I've been in my current relationship nearly two years. My current partner was in a polyamorous relationship prior to ours, and was seeing multiple partners when we got together. I tried to do that in the beginning but couldn't. We decided to become monogamous. Over the next year I struggled with why I couldn't, and realized that it has more to do with my own insecurities and attachment/abandonment issues than it does anything that she's done so far. I was jealous of everything: the fact that she had so many facebook friends, the fact that so many people text her, the fact that she can wear dresses and look better than me (and possibly get attention from other people), the fact that she's still friends with (and talks to on a regular basis) previous sexual and relationship partners....the first year of our relationship was HELL. And I scrambled to do the work on myself that I needed to do because she was not doing anything. It was all me. It was all what I had previously thought of these things that was 'wrong'. And I recognized that within myself: I wanted to change.
So I did everything I could. Counselling, talking, you name it. And then sometime maybe around a year or just prior to a year of being together I noticed one day that it was gone. All those feelings, the jealousy, envy, insecurity, mistrust, gone. I'm not perfect. I still experience twinges of it. But I think what I needed was to a.) see that she really cared about me and that I could trust her, which all her actions showed (and continue to show), and b.) give myself time to fully process the reasons that I felt the way I did about what I considered to be taboos and 'cheating'. And to be honest, I am surprised with how open-minded I actually am, underneath all of the stuff that was sitting on top of it for so many years.
So now, I don't care if she watches porn. Sometimes we watch it together. I don't really care if she flirts with others. I occasionally flirt with someone but we still have the arrangement of being monogamous so I honor that, and I know she does too. I don't need to check her phone and computer and email to know that. I just trust her, which is something I didn't ever think I'd be able to do. She chats constantly with people online. I have to admit that when I do feel twinges, it's around this. But I have confidence that it will fade over time, as it already is fading.
I sincerely believe it is up to the individual. Nobody can tell you what is cheating and what is not. Every partnership is different. It all comes down to what you agree to honor in your relationship. And then to talk about what happens if one of you crosses a line and breaks one of the agreements. Because that happens too. And you'd be surprised sometimes who ends up doing the crossing. It can happen to anyone, really. Not saying that we're not each responsible for our own behavior, but the exact opposite in fact.
I know a lot of what I've outlined doesn't have much to do directly with your situation. I guess I'm just trying to outline that the most important thing is that the agreement is verbalized, and that each person take ownership for their feelings and actions.
Hope it helps.