What should I consider if my boyfriend cheated on his ex two years ago?

I've been in a relationship with my boyfriend for over a year now, and he’s my first serious relationship, just as I am his. I'm an 18-year-old female, and he's an 18-year-old male. About ten months into our relationship, I used his iPad and found some concerning posts on his social media feed. As I explored further, I noticed he had been liking several suggestive pictures of attractive girls, including some recent ones from just a few days prior. This behavior was surprising since he seemed more discreet on his phone, but it turned out he had a separate account on the iPad. I confronted him about it, and since then, he has stopped using the iPad and given me access to all his other social media accounts. He’s been very patient as I work through my feelings and acknowledges the impact his actions had on our relationship. He’s genuinely trying to make amends. I’ve been gradually forgiving him because he hasn’t cheated on me, right? However, earlier today, I found myself bored and went through some of his old messages. I came across a conversation with a female friend from two years ago, which included explicit messages between them. I checked the dates, and they coincided with the time he was still involved with his ex-girlfriend. He would turn to her whenever he had issues with that relationship, and it seems they were never caught during that time. Although they haven't communicated in over a year, I’m struggling with how to process this new information. I'm a firm believer that once a cheater, always a cheater, yet this was two years ago, and he doesn’t talk to anyone else besides me and his family now. We spend nearly all our time together, and he effectively lives with me, so I’m aware that he doesn’t have much of a social life outside our relationship. Despite how it may sound, he has been devoted to me since I discovered he was liking and engaging with other women online. I occasionally check his accounts, and everything seems clean. I regret looking into his past, but I can't shake the feeling that it may hint at future issues. It’s frustrating because I see that he’s making an effort to change, and he has been understanding of my feelings. If we continue, I know it would be difficult for him to betray my trust since I have access to everything and am aware of his every move. I’m unsure how to approach this situation or what to think about it all. What should I do next?