I don't understand it at all, she loves me and acts like she's my girlfriend yet she doesn't want to be my girlfriend.
This really reminds me of my ex boyfriend, Dominic. I'm sure I've mentioned it before, but towards the end of our relationship (the third, FINAL break-up) he still wanted to be with me, but in public, he wouldn't even show that we were still somewhat together. In a group of friends he wouldn't want to be seen with his arm around me, but when we were alone in his room he'd ask me to have sex. Yeah I ended that shit right there and then, I couldn't believe his audacity.
Point is, although your case seems far less severe and the gender roles are switched, perhaps in Lindsey's mind, she can see you more often since you're just a friend, and not like, the "social desegregated" boyfriend? What I mean by that is like, she doesn't have to think about you as THE BOYFRIEND outside of her circle of friends, and perhaps her parents will think the same way. If she says she still loves you, it may not be entirely that she's worried about you not committing to her (although she could be a little bitter about you mainly breaking up with her for things that were mostly out of her control), but that really isn't a huge issue to squabble over for too long. Honestly, if you two are acting as if you're still dating, and at least stay exclusive with one another in those regards, I wouldn't necessarily (for now!) complain about how she doesn't want to be "your girlfriend."
I think that you're more worried about the label itself, so that your relationship bears some sort of seal of legitimacy. Although I'm not telling you to just backseat this whole issue, enjoy the time you can share with her now, doing the things you can do with her. However, if it goes on for a while without her talking/making any effort to say that she'd like to be your girlfriend because she indeed loves you, I'd say put your foot down. Labels aren't everything, but you don't need a label for everyone else. You don't even need it if can just be discussed/understood that you two love each other, and don't want anyone else in the way that you want each other. Sigh, I dunno. GIRLS ARE FICKLE THINGS, but if you feel the need to, just tell her that you can either be just friends, or a pair of people in romantic love. She can't have her cake and eat it too! (as in acting as if she's in a relationship with you but not willing to believe that she is)