I'm with this guy, but we haven't defined our relationship yet. What would you say we are?

It sounds like one of two things. One: he's afraid of commitment and therefore will offer none. Two: he wants to keep his options open. That way, if he decides to go with someone else, technically he didn't do anything wrong because you have not agreed to exclusivity. So right now I'd say you're friends with benefits, which usually turns out to be an unhealthy situation, since one person almost always develops feelings for the other that are not reciprocated.