While reading through Rabbenu Bachya, I noticed an interesting comment. He notes the shift from the description of Eliezer from "servant" to "man," beginning in Bereshit 24:21, and states that it coincides with the meeting up with/intervention of the mal`ach.
I would suggest two other reasons for this sudden shift.
1. Until this point, he has been following the directives of Avraham, and is indeed Avraham's servant. However, once interacting with other people, and negotiating on Avraham's behalf, he must do so from a position of strength. Therefore, despite introducing himself to them as the servant of Avraham, he is described as "the man."
2. While we know that he is the servant of Avraham, the people in the city of Nachor, and specifically, Rivkah and family, have no idea who he is. To convey this lack of knowledge, he becomes simply "the man." This has a connection to the general loss of names in the entire narrative, including, for example, Rivkah, for similar purpose. I posted about this more general phenomenon in 2004 in the post "Was the servant of Avraham Eliezer?"