Why does Harry think of his mother as “Lily” and not as “his mother”?

Question

Rereading the sixth book I don't understand why Harry thinks of his mother by name:

Neville’s childhood had been blighted by Voldemort just as much as Harry’s had, but Neville had no idea how close he had come to having Harry’s destiny. The prophecy could have referred to either of them, yet, for his own inscrutable reasons, Voldemort had chosen to believe that Harry was the one meant. Had Voldemort chosen Neville, it would be Neville sitting opposite Harry bearing the lightning-shaped scar and the weight of the prophecy. . . . Or would it? Would Neville’s mother have died to save him, as Lily had died for Harry? Surely she would. . . . But what if she had been unable to stand between her son and Voldemort? Would there then have been no “Chosen One” at all? An empty seat where Neville now sat and a scarless Harry who would have been kissed good-bye by his own mother, not Ron’s?

Answer 1

In this instance, this is a literary perspective known as Third Person Limited

In third-person limited narration, the narrator still exists outside the events of the story, but does not know the motivations or thoughts of all the characters. Rather, one character is the driver of the story, and the reader is given a closer peek into that character’s psyche than the others.

We're given a peek into Harry's thought process, but it's the narrator that is narrating what they see in Harry's mind rather than Harry narrating their own thoughts. As such, the narrator would know her as Lily rather than 'Mum'