[jimex289]

There's Ayumi Hamasaki/Ayu the person and Ayumi Hamasaki/Ayu the singer. Early in her career she made it a point to distinct the two. She wanted to separate them because fans were only seeing her as Ayumi Hamasaki the popstar and not as Ayumi Hamasaki the person/the "real and true" Ayu. She still continues to do so because she believes they are not the same but with the idea that each has a direct bearing/effect on the other. I think it's understandable. If I were her I wouldn't want to mix my personal/private life with my career as a singer. Though she wants to hold this distinction, I read that even when she's not singing or working and out for personal business, if she encounters a fan, she is no longer the "real" Ayu but Ayumi Hamasaki the popstar; she believes this is her responsibility to her fans. Though they are "different people", they are really still two faces on the same quarter. Those "faces" flip depending on the situation. That's the kind of thing she accepted when she became and singer and she knew full well what she was getting herself into.

If you want a more simple response: Ayu is just a nickname. She's had it even before she debuted. It refers to none other than Ayumi Hamasaki.