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

River Flows in You por Yiruma was a fã picked lullaby. That means, before all the filmes 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.