Another comment said you can safely assume he’s your boyfriend. Before I did that, I’d want to have a clear and explicit conversation about exclusivity (do you two want to have it or not?).

As to your broader question, sure the label to the relationship might help in clarifying expectations.

But more important is what kind of person you want to be to him and vice versa. Years from now, if you were to look back on what kind of person you were in this period, what would you like to see? Kindness? Exploration? Consistency? Honesty? Playfulness?