ok i am really confused! The scene in twilght where edward is playing on the piano next to bella, is that bellas lullaby or is the song the river flows by you the lullaby?
yes. That lullaby playing in the scene where Edward is playing for Bella, that is the lullaby. It's official name is Bella's Lullaby by Carter Burwell.
River Flows in You by Yiruma was a fan picked lullaby. That means, before all the movies came out, everyone liked that and wanted the lullaby to resemble that song.
New Moon, there was a completely different style of score music, instrumental pieces. There was no lullaby in New Moon.