How did the Unspeakables know to change the records in the Hall of Prophecy?

Question

Related: How do you listen to a prophecy without smashing it in Harry Potter?; How do the spheres with recorded prophecies appear in the Department of Mysteries?

We know that the contents of the prophecy concerning Harry were not common knowledge, and we know that neither Rookwood, who was an Unspeakable working in the Department of Mysteries before Voldemort's downfall, nor Bode, who was an Unspeakable placed under the Imperius Curse by the Death Eaters, could access the prophecy's contents, or else Voldemort would have been saved the trouble.

But at the end of Book 5, Dumbledore tells Harry this:

"The official record was relabeled after Voldemort’s attack on you as a child," said Dumbledore. "It seemed plain to the keeper of the Hall of Prophecy that Voldemort could only have tried to kill you because he knew you to be the one to whom Sibyll was referring."

So how did this "keeper" guy know to relabel the prophecy with Harry's initials, anyway, if he couldn't even get at its contents?

(Speculation is welcome if you have sound logical support.)

Answer 1

I suspect they simply listened to it

Dumbledore does indeed say that only the subjects of a prophecy can remove it from the shelf.

`And then you saw Rockwood, who worked in the Department of Mysteries before his arrest, telling Voldemort what we had known all along -that the prophecies held in the Ministry of Magic are heavily protected. Only the people to whom they refer can lift them from the shelves without suffering madness....'

—Harry Potter and the Order of the Phoenix

However, there is no reason to interpret this quote as implying that, from the very moment the prophecy is recorded, it is thus protected. The protections placed on prophecy records could easily be placed only after the Keeper has listened to them. Alternately, while the spells placed on the prophecies may automatically appear once the prophecies are recorded, it may be possible to remove them with enough effort. If Voldemort would have to stand in front of the prophecy for days, chanting, it would have been essentially impossible to remove the protections.

If so, why not kidnap the Keeper?

Why not target the Keeper, in either case? As someone who has listened to Harry's prophecy, there's a decent chance that they still remember the details, in which case Voldemort could simply kidnap them. It is possible that there is some additional security involved, such as Memory Charms cast on the Keeper. But I think there are two much simpler possibilities.

1. Voldemort wished to avoid drawing attention to himself. The disappearance of one who was surely a top Ministry official would do exactly the opposite, particularly if this happened after Bode attempted to steal the prophecy. Voldemort perhaps considered it, but considering the distinct possibility that the Keeper did not remember the details, why not wait a few months and tempt Harry Potter into removing it for him?
2. The Keeper was dead. Simple as that. The person who had listened to and relabeled Harry's prophecy was no more, and the only remaining record was on a shelf in the Hall of Prophecy. Of course, there would have been another Keeper, but they wouldn't know what the prophecy contained.

Answer 2

Theory 1:

This could be completely off target. Some people believed Voldemort was dead. I am saying this based on

Hagrid's View:

Some say he died. Codswallop, in my opinion. Dunno if he had enough human left in him to die. Some say he’s still out there, bidin’ his time, like, but I don’ believe it. People who was on his side came back ter ours.

_{^{Harry Potter and the Sorcerer's Stone, Chapter 4, The Keeper of the Keys}}

Snape's View:

“You ask why I did not attempt to find him when he vanished. For the same reason that Avery, Yaxley, the Carrows, Greyback, Lucius”— he inclined his head slightly to Narcissa — “and many others did not attempt to find him. I believed him finished. I am not proud of it, I was wrong, but there it is... If he had not forgiven we who lost faith at that time, he would have very few followers left.”

_{^{Harry Potter and the Half-Blood Prince, Chapter 2, Spinner’s End}}

Keeper might have assumed that Voldemort was killed because Harry survived and Voldemort disappeared.

Theory 2:

There were two possible candidates to rival Voldemort, Harry and Neville.

“The odd thing, Harry,” he said softly, “is that it may not have meant you at all. Sibyll’s prophecy could have applied to two wizard boys, both born at the end of July that year, both of whom had parents in the Order of the Phoenix, both sets of parents having narrowly escaped Voldemort three times. One, of course, was you. The other was Neville Longbottom.”

_{^{Harry Potter and the Order of the Phoenix, Chapter 37, The Lost Prophecy}}

There were very few families who survived attacks by Voldemort but Harry not only survived the attack he managed to defeat Voldemort.

I should remind you that when Potter first arrived at Hogwarts there were still many stories circulating about him, rumors that he himself was a great Dark wizard, which was how he had survived the Dark Lord’s attack. Indeed, many of the Dark Lords old followers thought Potter might be a standard around which we could all rally once more. I was curious, I admit it, and not at all inclined to murder him the moment he set fool in the castle.

_{^{Harry Potter and the Half-Blood Prince, Chapter 2, Spinner’s End}}

The keeper might have assumed Harry was the wizard born to challenge Voldemort.