ok i am really confused! The scene in twilght where edward is playing on the Piano Weiter to bella, is that bellas lullaby oder is the song the river flows Von Du 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 Von Carter Burwell.

River Flows in Du Von Yiruma was a Fan picked lullaby. That means, before all the Filme 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.