Are we dating? What do I do and say to him?

So there's this guy I've been seeing for the past month and a half, and we've known each other for a few months now because we worked together over the summer. Since we've been back at college we've been spending a lot of time together. About a month ago I called him up and asked him what we were and his response was essentially he just got out of a long-term relationship over the summer, and isn't ready to date yet but thinks we're really good friends, wants to get to know each other better, and wants to see where this goes. I indicated to him I was past the part of my college life where I was interested in hook ups and friends with benefits, and he immediately apologized saying that is not what he wanted from me at all.

Since then we've continued to spend a lot of time together, and he's kissed me in public, I went on an overnight trip with him, we've slept in the same bed multiple times, (not actually slept together though) and have spent significant time together and in public. We also text each other everyday, all the time. He's talked about visiting me at home over fall break, me visiting him when he studies abroad, and seeing each other over the summer because it is likely we will be working in different cities. He's also talked about meeting my parents and has indicated to me that he is not seeing other people. However, he has not explicitly said what our future entails or where we currently stand.

Considering all the developments since the last time I asked him, is it fair for me to ask him again? Or should I wait and see where it goes?

Updates
+1 y
We have not defined the relationship, but is it fair for me to ask that we do?
Updates
+1 y
I asked him about it yesterday and he basically said he wasn’t looking for anything serious but really likes me and doesn’t want to stop talking to me. He said he wasn’t seeing anyone else, but also did not promise exclusivity. I don’t want to stop seeing him but I also don’t want to be in a situation where I get hurt. What do i do?
Are we dating? What do I do and say to him?
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