Since, motion of the ball on the beam depends on the beam angle alpha, shouldn't this system have a single degree of freedom?
Try to think of degrees of freedom as objects that can move independently of each other. They may be coupled like in your system such as alpha and theta so you can describe one of them with the other... however the ball is not. I would suggest "the art of modeling in science and engineering with mathematica" https://www.amazon.com/Modeling-Science-Engineering-Mathematica-Second/dp/1584884606
If you'd like a good read.