ok i am really confused! The scene in twilght where edward is playing on the piano volgende to bella, is that bellas lullaby of is the song the river flows door u 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 door Carter Burwell.

River Flows in u door Yiruma was a fan picked lullaby. That means, before all the films 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.

Of course the song in the movie. It´s on the soundtrack too.