Dan is using your second example. It's already been said that it was nice to meet Dan, now Dan replies with a different stress, simply to avoid a robotic response. He is stressing that it is indeed a pleasure to meet Lee.
Think of it this way. You have just been introduced to Nelson Mandella, or someone else who is your personal hero. It was accidental, and Mr Mandella has been going down a reception line, repeating like a robot, "Nice to meet you." "Nice to meet you." "Nice to meet you." Then he stops, reads your nametag, extends his hand, and politely says, "Nice to meet you, Sel. I've heard about you."
If you could prevent yourself from fainting dead away at this point, and find it possible to speak at all, you would say, "Nice to MEET YOU, Mr Mandela" and then blush very brightly.
In your example, Dan wasn't all that excited to meet Lee, but he did want to stress what comes across as at least equal pleausure.