"Nice to meet you, I am (I'm)...." or "Nice to meet you, my name is" What reads better....or....who cares?
Depends upon the character and the scene. "I'm Jim Bond" is fairly informal. "My name is James Bond" is fairly formal. "The name's Bond. James Bond." is quite distant and pompous. I'd just started at a new place and the MD comes in, offers a handshake and introduces himself simply as "Trevor Jones". I later found out he was Phillip Trevor-Jones...
Muahahaha... Right.... So, I'll mix it up between I'm and My name.... just so its not the same... or I'll keep it "I'm" for the MC. She is fairly open and informal with everyone she meets. In fact, too informal, so that may be the way to go. She has been corrected by another character based on her behavior already in Chapter 3. Nothing major, she just interrupted a Doctor while he was talking... a pet peeve of his, and he let her know it.... Moments later he interrupted her and what did she do? Corrected him... She is brave.