Why didn't Severus save this character?

In the Harry Potter and the Deathly Hallows - Part 2 film adaptation we see Severus Snape crying over the dead body of Lily Potter. In this scene we can see that Harry is in the same room.

In the very beginning of the books, we are told that Harry was saved by Rubeus Hagrid:

‘No, sir – house was almost destroyed but I got him out all right before the Muggles started swarmin’ around. He fell asleep as we was flyin’ over Bristol.’

Harry Potter and the Philosopher's Stone, Chapter 1: The Boy who Lived.

I haven't read the book in a while and my memory is not flawless, but I suspect the scene above doesn't happen this exact way in the book. If so,

1. What does actually happen? (book canon).


2. How is it that Severus wasn't the one who saved Harry Potter? (film canon).