To me, the fishing for that conversation seems to be another test to see where you are at. To see if you are still pining for her, bending over backwards for her, etc. etc.
One thing I've talked to a friend of mine about and one thing I definately agree with is that if a girl wants to be completely done with you, they will be. They wouldn't want to come back to you for little things that are seemingly irrelevant. My ex called me two weeks after she dumped me to ask for her brother's video game. Not only could he have done it, the fact is she called to ask for it back. I'm not saying read into it like it's a positive sign of getting back together, but see it for what it is.
The fact she is expressing to you that she has already moved on is in my mind complete bullshit. The thing about proving something is that if you are really confident in how you feel, you will not feel the need to bring it up and make a big deal of it. You will be satisfied with it personally and it won't be necessary to bring it up to others. I don't buy it if she has to tell you she has.
As for still contacting you after the breakup talking about gifts, you need to let her know that it's not okay to do that. I'm assuming she dumped you, and the fact that she is bringing it up and bringing up your past, it's not cool at all. You are trying to move on and she is bringing up old issues that you are trying to come to peace with. Not only is that selfish, but it's not getting either of you anywhere.
Waking up next to a beautiful girl,
Step outside and say hello to my beautiful world.