When she asks for your full name, it is because they want your name for screening purposes. A fake name serves no purpose in that reguard.
I never delete any of my messages. Those i haven't seen, remain numbers in my phone. Those i have seen are given nicknames i choose to give them as a reminder of how the visit went. (I am bad with names, and names meen nothing if you have a list of 40 Mohammed's, 8 David's, 10 Jason's, 20 Adam's and a long list of names you can't even pronounce.) So nicknames that refer to the time spent jog my memory instantly about if I enjoyed time with the person, and am I willing to return and see them again, as if I didn't enjoy my time with them, I refuse to return.