Your lover doesn’t introduce you to people as his lover because that would be really weird and kind of creepy and totally awkward for everyone because it’s basically announcing that the scope of your relationship is purely, or at the very least mainly, sexual and nobody needs or wants to know all that. And since the scope of your relationship is purely, or at least mainly, sexual, you aren’t boyfriend and girlfriend no matter how many weekend trips you’ve been on or how many friends and family members you’ve met or how often you talk to each other. “Open relationship” and “non-exclusive” are concession titles. They give you the “relationship” you want while granting him the no-strings-attachment that he wants.
You need to be honest with yourself about what kind of relationship you really want and if it’s more than regular sex with someone you like (even someone you care very much about). If you want to be someone’s actual girlfriend, then you also need to be honest with yourself that this isn’t the relationship for you. This man knows what he wants. He’s likely been around the block quite a few times by now and isn’t interested in a committed, monogamous relationship with you. He’s confident enough in that choice, and in what you two share together, that he has absolutely no need or desire to define your relationship and what you mean to each other to anyone else. You, on the other hand, are less confident in what you share together and what your roles are in each other’s lives. And you need to ask yourself why. If you are truly OK with just being his lover, you wouldn’t need or want that title given during casual introductions.
If you have a relationship/dating question I can help answer, you can send me your letters at firstname.lastname@example.org.