Well, based on what you've said, sounds like you are his girlfriend by most people's definition, if his family know about you etc. However he seems to make it clear that he doesn't want the commitment and expectations that come with a relationship, and the label.
Honestly it doesn't sound great and like you'll get the life/ relationship you want with this person. Do you really really love him and that's why you put up with it/ hope things will change?
I have an ex that doesn't sound too dissimilar in some ways. We had a very detached relationship that made me sad even though we loved each other so eventually I just ended it, and in my head was finally ready to move on. He then went on a charm offensive, gifts, contacting me daily, proper open vulnerable conversations about his feelings, offering things that were off-limits before. I was single so gave him the benefit of the doubt while emotionally detached from it all.
Guess what? Past the first few weeks of bonding and amazing sex, nothing had changed.
He may well have feelings for you, but he wants a casual relationship with the occasional emotional and physical echange that comes with it, he wants someone in his life that's there for him with low effort, almost like a low maintenance house plant. And rekindling with an ex is the easiest way to achieve this.
You on the other hand want a proper loving relationship with all the bells and whistles, and it sounds like you won't get there with him. You want someone that tells you they love them, not someone that says 'you're not my girlfriend'. IME people don't change quite as dramatically, especially if he has already promised you the world and gone back on it after you said yes.