Harry learns things that only Dumbledore would have known.

Does he? It certainly seems possible that Harry is just filling in the blanks himself. I just went back and re-read it. Consider:

"Explain," said Harry.
"But you already know," said Dumbledore. . .
...
"But if Voldemort used the Killing Curse," Harry started again, "and nobody died for me this time -- how can I be alive?"
"I think you know," said Dumbledore. "Think back. . ."

The information that Dumbledore actually does provide to Harry is either inconclusive or insubstantial -- e.g. Harry asks about the peculiar behavior of his wand, and Dumbledore says he cannot but guess. Harry asks where they are, Dumbledore cannot answer and says that they are where ever Harry thinks they are. Harry asks about the Deathly Hallows:

"Real, and dangerous, and a lure for fools," said Dumbledore. "And I was such a fool But you know, don't you? I have no secrets from you anymore. You know."
. . .
So you'd given up looking for the Hallows when you saw the Cloak?"
"Oh yes," said Dumbledore faintly. . . "You know what happened. You know."

Dumbledore tells Harry of his stint with Grindelwald, but Harry might be piecing together the narrative from what he has learned (Dumbledore again seems to indicate at some times that Harry already knows what happened).

The chapter's ending could go either way:

"Tell me one last thing," said Harry. "Is this real? Or has this been happening inside my head?"
...
"Of course it is happening inside your head, Harry, but why on earth should that mean that it is not real?"

This all leads me to consider that chapter very shaky evidence. I'd still say the strongest evidence is the ghosts. Eliezer's explanation is lacking, since in Book 4 a group of travelling ghosts visits Hogwarts from elsewhere beyond the castle (the gang of headless ghosts, entry into which Nearly-Headless-Nick is sadly not quite eligible). Myrtle once left the castle to haunt a wedding. So, in canon, ghosts must be more than just after-images bound to the castle stone. Though there are, of course, other similar explanations. . .