@tee95,
I had some additional thoughts after reading all the posts again.
You asked: "Should I forgive my boyfriend and how?"
I think that you already have forgiven him. This is based on you searching for answers like on this forum. However, what I think you really meant to ask, is "how do I go back to trusting him like I did before seeing that chat". I think your natural response is the answer-- the "snooping".
You forgave him, but that doesn't change the effect that chat had on your trust. You have solid reason to look more closely at more personal aspects of his life for more indication of synonymous activity. The checking his life more closely, is only a temporary thing. Your trust for him has been wounded, and the only way to heal it back to homeostasis (regular working function) is to satisfy the questions in your mind that require some sort of "proofs" in order to put back together what that chat broke.
The absolute most important thing on your behalf from my perspective, is to be totally upfront and honest. The worst thing you can do, is to sneak around, lie, or steal things from him. Why? Because in doing so, you are giving him solid reason to doubt your trust and character. You are telling him lies, stealing from him, and sneaking around like a spy. You may feel, "its the only way to get the absolute truth", but in doing so, you are compromising your own values in the relationship. If you start hiding things from him now, its possible to become a habit that he may not cherish in his future wife.
Trust is one of the coolest aspects of relationships I think. It is something that has to be 100 percent earned. You can't just give trust or take it away, you need actual solid evidence to sway trust in one way or another. You can lie about trusting someone, but you can't lie to yourself about the doubts that are bouncing back and forth in your own mind. Your quest to rebuild that trust for him, is resolved through your actions AND the variable of time. As time progresses, and you find only reasons to trust him more, you will again be at that blissful state of not having any reason to doubt his commitments to you.
As for why that chat may have happened. There is the possibility that he felt insecure about you being on vacation in Thailand for 20 days. He may have had doubts in his mind that you were playing him for the fool. Guys aren't as open to share their fears as girls are. Yet we all know, most people on vacation usually have a more leisurely state of mind, are more prone to drinking, and more likely to meet with other strangers looking for "a good time". There is always a sense of uncomfortableness that you never know who the one you love might meet when hey are all alone on a vacation. I'm sure if there were statistics, they would show affairs, or infidelity has the highest occurrence of happening on Vacation hot spots. This is due to the nature of distance that will make it less likely that this person will ever cross paths in your regular life. Making it less likely to ever have any evidence besides your loved one's own words, which they can easily lie about.
I do get the sense that he cares for you a lot. In this case, wanting to know if something happened, I think taking his word for it would be best. This is based upon the fact that this is the first time in 2.5 years that he has done anything that gave you reason to doubt him. He has earned this privilege based on his past with you. That said, you still want to get back to the full trust you had for him. Ask to see anything he has that will help restore your trust (his phone, e-mails, chat friends, etc). For things you can't prove that are in the past, his word is worth accepting, because you won't ever be able to know for sure. Even if he lied to you and said "YES, we did do this", you still don't know if he is being honest, or if he is just sick of being questioned for something he already answered.
Being 23, and knowing that marriage means "Till death do us part". I believe you have to have that 100 percent trust after 2.5 years if you are planning on making promises to be with each other for at least another 50 years.
I hope these thoughts help you with your cognitive process of doing the most right thing. This is your relationship and nobody knows what to do in your shoes better than the person walking in them.