When/How did Snape become a double agent?

Question

Judging from the quote below, was Snape already a double agent when he found out about the meaning of the prophecy?

'Well, Severus? What message does Lord Voldemort have for me?

'No - No message - I'm here on my own account! ... 'I - I come with a warning - no, a request - please -' ... 'What request could a Death Eater make of me?' 'The - the prophecy... the prediction... Trelawney...'

'Ah, yes,' said Dumbledore. 'How much did you relay to Lord Voldemort?'

'Everything - everything I heard!' said Snape. 'That is why - it is for that reason - he thinks it means Lily Evans!'

'The prophecy does not refer to a woman,' said Dumbledore. 'It spoke of a boy born at the end of July -'

'You know what I mean! He thinks it means her son, he is going to hunt her down - kill them all -'

'If she means so much to you,' said Dumbledore, 'surely Lord Voldemort will spare her? Could you not ask for mercy for the mother, in exchange for the son?'

'I have - I have asked him -'

'You disgust me,' said Dumbledore, and Harry had never heard so much contempt in his voice..

This quote seems to imply that Dumbledore was:

• Not surprised to see Snape
• Had given Snape information to relay to Voldemort

However, the general consensus (from what I remember) is that Snape became loyal to Dumbledore only after Lily's death. Given that both sides usually know they are dealing with a double-agent. How did Snape become one?

Option 1: Voldemort sent Snape to find information on Dumbledore.

Option 2: Snape went to Dumbledore to help save Lily.

Option 2 does not seem to fit based on what I take the quote to imply. So, if Option 1 is the case, do we know when it happened?

This question is not a duplicate of Was Severus Snape made a Death Eater before or after he told Voldemort about the Prophecy? because that question is about when Snape became a Death Eater not a double-agent.

Answer 1

Snape becomes a double agent moments after the quote you describe.

Context is key here, and from the sound of it you're considering the quote out of context. Snape and Dumbledore haven't met accidentally - it's a prearranged meeting by Snape.

The corridor dissolved, and the scene took a little longer to reform: Harry seemed to fly through shifting shapes and colors until his surroundings solidi- fied again and he stood on a hilltop, forlorn and cold in the darkness, the wind whistling through the branches of a few leafless trees. The adult Snape was panting, turning on the spot, his wand gripped tightly in his hand, waiting for something or for someone. . . His fear infected Harry too, even though he knew that he could not be harmed, and he looked over his shoulder, wondering what it was that Snape was waiting for—

Then a blinding, jagged jet of white light flew through the air. Harry thought of lightning, but Snape had dropped to his knees and his wand had flown out of his hand.

“Don’t kill me!”

“That was not my intention.”

Any sound of Dumbledore Apparating had been drowned by the sound of the wind in the branches. He stood before Snape with his robes whipping around him, and his face was illuminated from below in the light cast by his wand.

“Well, Severus? What message does Lord Voldemort have for me?”

Dumbledore assumes the meeting was to deliver a message from Voldemort, hence his disarming of Snape, and Snapes fear. They are not allies at this point.

Directly after this though, Snape explains he is there to warn Dumbledore that Lily is in danger, and begs him to keep her safe. He even explains that he has asked Voldemort not to kill her.

“You disgust me,” said Dumbledore, and Harry had never heard so much contempt in his voice. Snape seemed to shrink a little, “You do not care, then, about the deaths of her husband and child? They can die, as long as you have what you want?” Snape said nothing, but merely looked up at Dumbledore.

“Hide them all, then,” he croaked. “Keep her—them—safe. Please.”

“And what will you give me in return, Severus?”

“In—in return?” Snape gaped at Dumbledore, and Harry expected him to protest, but after a long moment he said, “Anything.”

Snape becomes a double agent on that last word.

Answer 2

Snape became a double agent, in theory if not in practice, when he begged Dumbledore to save Lily - several lines after the quote you have above:

"Hide them all, then," [Snape] croaked. "Keep her - them - safe. Please."

"And what will you give me in return, Severus?"

"In - in return?" Snape gaped at Dumbledore, and Harry expected him to protest, but after a long moment he said, "Anything."

This turned into a more concrete agreement to act as a double agent on the next page, in the next memory scene where Dumbledore and Snape talk after Lily's death:

"I wish... I wish I were dead...."

"And what use would that be to anyone?" said Dumbledore coldly. "If you loved Lily Evans, if you truly loved her, then your way forward is clear."

Snape seemed to peer through a haze of pain, and Dumbledore's words appeared to take a long time to reach him.

"What - what do you mean?"

"You know how and why she died. Make sure it was not in vain. Help me protect Lily's son."

"He does not need protection. The Dark Lord has gone -"

"The Dark Lord will return, and Harry Potter will be in terrible danger when he does."

There was a long pause, and slowly Snape regained control of himself, mastered his own breathing. At last he said, "Very well. Very well. But never - never tell, Dumbledore! This must be between us! Swear it! I cannot bear... especially Potter's son... I want your word!"

I think you are misreading the quote you've provided. Specifically, I have a different interpretation of these lines:

'The - the prophecy... the prediction... Trelawney...'

'Ah, yes,' said Dumbledore. 'How much did you relay to Lord Voldemort?'

I don't believe this indicates that Snape relayed information to Voldemort on Dumbledore's orders or behalf. To me, this indicates that Dumbledore either knew or quickly deduced that Snape had overheard or learned of the Prophecy, and that he would have told Voldemort in his normal role as a Death Eater.

Therefore, this snippet isn't indicative of an existing double relationship; it is indicative of the stresses that Dumbledore took advantage of to create one subsequently.

Follow-up based on the comment:

So only after Voldemort's "death" did Snape become a double-agent? Why was Dumbledore not more surprised to see Snape then? Only because Dumbledore is not one to show such an emotion?

Snape bit on the hook when he promised to do anything, and Dumbledore set the hook when Snape became unraveled after Lily's death. The fact that Dumbledore was impassive either means that he had, to some extent, predicted events, or that he was not showing emotion. To quote Aberforth:

"I knew my brother, Potter. He learned secrecy at our mother's knee. Secrets and lies, that's how we grew up, and Albus... he was a natural."