A lot of you are debating on wether the second song really IS Bella's Lullaby, and it most definately is. There were questions about using "River Flows In You", by Yurima, but since all of the other songs made especially for Twilight The Movie was made by Carter Burwell, they used his Bella's Lullaby, which HE made. So the song is CARTER BURWELL - BELLA'S LULLABY, and you'll find it on the album "CARTER BURWELL - TWILIGHT (THE SCORE). Thanks for reading, it will save yourself some time, not having to ask questions. Cause I'm really not anwering any more.
The movie starts in Edwards room and they're dancing, then they fly out in the woods and climb up trees while "bella's lullaby" is playing in the background. At the end Edward is playing it on the piano for Bella. If you've read the book, when Edward plays it for her, he says he got inspiration from her, and "Bella's Lullaby" was actually a song he was singing to her so she would fall asleep, but we didn't see this in the movie.