Why did Dumbledore say Snape saved Harry because James saved Snape?

Question

At the end of Philosopher's Stone, Dumbledore tells Harry that Snape was trying so hard to protect Harry because James once saved Snape's life.

He could have said the complete truth, something like:

Snape hates you because you look like your father, who was a bully who, together with Black, made Snape's life impossible and once almost killed him. However, Snape will do anything to protect you because he loved your mother.

Or, if that was supposed to be a secret, he could have said a white lie, only telling part of the truth:

Snape and your father didn't get along very well. But there is no way he would let any student die, if he can avoid it.

But why did he tell that lie?

Answer 1

Dumbledore never told Harry, because Snape did not wish him to.

Seen from this quote:

Professor Albus Dumbledore: [after James and Lilly's deaths] The boy survives.

Professor Severus Snape: He doesn't need protection, the Dark Lord is gone!

Professor Albus Dumbledore: The Dark Lord will return! And when he does, the boy will be in terrible danger! He has her eyes. [Snape pauses in shock]

Professor Albus Dumbledore: If you truly loved her...

Professor Severus Snape: No one... can know.

Professor Albus Dumbledore: That I shall never reveal the best of you, Severus?

And this quote from the same site:

Severus Snape: You're just like your father. Lazy, arrogant...

Harry Potter: Don't say a word against my father!

Severus Snape: - Weak.

Harry Potter: I'm not weak!

Severus Snape: Then prove it!

If I am not mistaken this is also in Deathly Hallows or Half Blood Prince.

As seen from the 5th book when Harry looks in Snape's pensieve, Snape does not want anyone to know his past or at least he doesn't want Harry to. He appears to want to suffer alone.

He also views Harry as the same as his father, as is explained through memories that he does everything that he does for Lily. That may be the reason why he wants no one to know his past.

I think Snape didn't particularly like the fact that Dumbledore said that the reason that he saved Harry was that James saved him, because from the scene that it shows Snape being "saved", he viewed it as James covering himself. Also, he couldn't tell the true reason that he did it because someone would find out about his cover and him avenging(possibly) Lily's death would have been not complete. So since the "truth" Dumbledore told was already mildly known he let it slip. He probably expected Harry to be haughty and rub it in face like his father.

Answer 2

Because James did save Snape’s life.

Dumbledore isn’t lying; there’s an incident described at the end of the third book in which Snape’s life really was saved by James:

“Severus was very interested in where I went every month.” Lupin told Harry, Ron, and Hermione. “We were in the same year, you know, and we — er — didn’t like each other very much. He especially disliked James. Jealous, I think, of James’s talent on the Quidditch field… anyway Snape had seen me crossing the grounds with Madam Pomfrey one evening as she led me toward the Whomping Willow to transform. Sirius thought it would be — er — amusing, to tell Snape all he had to do was prod the knot on the tree trunk with a long stick, and he’d be able to get in after me. Well, of course, Snape tried it — if he’d got as far as this house, he’d have met a fully grown werewolf — but your father, who’d heard what Sirius had done, went after Snape and pulled him back, at great risk to his life…. Snape glimpsed me, though, at the end of the tunnel. He was forbidden by Dumbledore to tell anybody, but from that time on he knew what I was…”

— Prisoner of Azkaban, chapter 18 (Moony, Wormtail, Padfoot, and Prongs)

And this is the passage you’re referring to when Harry is in the hospital wing, recovering from his encounter with Quirrell:

“Quirrell said [Snape] hates me because he hated my father. Is that true?”

“Well, they did rather detest each other. Not unlike yourself and Mr Malfoy. And then, your father did something Snape could never forgive.”

“What?”

“He saved his life.”

“What?”

“Yes…” said Dumbledore dreamily. “Funny, the way people’s minds work, isn’t it? Professor Snape couldn’t bear being in your father’s debt… I do believe he worked so hard to protect you this year because he felt that would make him and your father quits. Then he could go back to hating your father’s memory in peace…”

Harry tried to understand this but it made his head pound, so he stopped.

— Philosopher's Stone, chapter 17 (The Man with Two Faces)

While Snape may have had other motivations for helping Harry (such as his promise to Dumbledore, or his love for Lily), there’s nothing to say that he didn’t also want to settle his debt with James. So it’s fine for Dumbledore to say this, even if it somewhat confuses Harry.