To be honest it sounds like you need to see where each of your priorities lie for the future together - if there is one, as she seems to be closed to you two being together "publicly". This makes me wonder if it's also to keep things open to the other partners or exes? I can't say what motives are, but it wouldn't sound good to me (my current partner began our relationship saying one day she might stray, I told her clearly for me it would be over if that happens).
I don't know, to me it just seems controlling, keeping you for what she wants and denying what she doesn't want. If you don't clear the air about what you want and get her to clarify her side, she'll just continue using you.