Boyfriend was sending messages to another girl and went out. Should I forgive?

Hello! I just found that my boyfriend was chatting with another girl when I wasn’t in town, by the way it’s my first time when I checked out his messages. We’ve been together for 1.5, I’m so broke now. He cried and hardly apologized. I don’t know what to do. He swears that he doesn’t have sex with her, even kiss. But their messages in front of my eyes now, he called her “my baby”, she called him “my love”, it was in last message and another day when I came in town he didn’t text her. I didn’t see all messages, but I also saw that she spoked about sex with him, but he wasn’t really enthusiastic about it. And when he cried and apologized he told that they went out one time for a coffee, that’s it.
My questions are:
1). What is it? I always thought that when you love someone you won’t even text like this with someone else.
2). Is it really cheating? He was swearing that it was just nothing. But for me it was really hurtful, because when I’ve been in another town he was talking with me so cold and bad. And then when I came I found out this!
Thank you for your attention, guys!
Boyfriend was sending messages to another girl and went out. Should I forgive?
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