My own personal take on the matter:
I think Voldemort found Snape’s request amusing. And I think that’s why he said he’d spare Lily.
Think about it - Snape’s a couple years out of school, he’s just some jumped-up little stripling of a baby Death Eater. And now he’s asking for a boon, for a favor. It’s hilarious.
But this dirt-poor halfblood must have brass balls to even ask him, and that’s amusing. And after all, Snape gave him the prophecy (or part of it, anyway) and Voldemort is certain now that he can destroy the only credible threat to his power - a helpless infant - and he’s in a good mood. So, he indulges Snape’s request. A little joke.
So, with Voldemort’s word that Lily would be spared, why did Snape suddenly bolt to Dumbledore, his enemy? Why would he risk his very life by going to a man who has every reason to kill him on sight without question?
Firstly, I think Snape was already second-guessing his decision to join Voldemort in the first place, possibly very soon after he finished school and joined the Death Eaters, if not immediately.
Keep in mind Snape’s original motivations with the Death Eaters. He was drawn to the junior Death Eaters in school because he felt powerless, victimized and disrespected. He’s recruited right out of the gate at the age of eleven by Lucius Malfoy, among others. They promise him power, they promise him respect, that he’ll never be that little boy crying in the corner while his muggle father rages over him, or that skinny unpopular kid being dangled upside down and having his pants pulled off by the popular jocks in school ever again. That’s what he was promised.
What did he get? A master. A charismatic “lord” who demands absolute obedience and unquestioning loyalty and who will punish severely anyone who doesn’t toe the line (e.g. Lucius Malfoy after OotP). He’s suddenly in with a pack of back-stabbing jackals lead by a ravenous serpent and all his dreams and ambitions are so much ash.
I think he figures out pretty quickly that Voldemort is in it for Voldemort, and while some are perfectly content to ride his coattails and pick up scraps, Snape - stubborn, prideful, young Severus Snape - is not. But he’s stuck.
So even before the prophecy & Lily being in danger, his own supposed loyalty to Voldemort, even at that point, I think is pretty thin stuff. He was never really a “true believer” in Voldemort or his ideology, not like Bellatrix or the Malfoys, I think, and so I feel like eventually reality just hit him in the face like a sack of wet rags. Not so much a crisis of conscience - that came later with Lily - but a raging case of Buyer’s Remorse, as it were.
But now Lily is in mortal danger and first Snape asks for her life, because he knows it’s absurd but he’ll try anything. Snape knows Voldemort cannot be trusted to keep promises because Voldemort is loyal to no one but himself. And Lily? She doesn’t stand a chance if he should change his mind on a whim. So regardless of what Voldemort said to him, he panics and goes to the only person alive who is powerful enough to defy Voldemort, no matter the risks or cost to Snape himself, because Lily is the only person who was ever kind to him and even though they haven’t spoken in years, he can’t stand the thought of living in a world she does not.
As for why Voldemort actually kept his word and told Lily to stand aside, giving her the chance to live - I think that also amused him. He did not understand love. He found the entire concept laughable and contemptible and I think he took a sick joy in watching his enemies beg, and in watching Lily beg for her son’s life. I think on some level he was also morbidly fascinated by the idea - the ridiculous notion that someone might be willing to die for their child. He probably did not actually expect her to refuse, to willingly give up her own life for someone else (how ridiculous!). More fool him….
