Bella's Lullaby Questions!? I Need Answers Please !! :]?

i thought that Bella's Lullaby was supposed to be "The River Flows In You" By Yuma. But when i went to see the movie it wasn't? i goggled it and it kept saying it was like a song by Carter Bur well. Why did they change it. OR did they even change it at all, like, was the one by Yuma just a big Rumor. and then i heard that Rob made his own lullaby. Why didn't they put that in?