I always see it is conventional (in Landau-Lifshitz for example) to decompose the angular velocity vector to 2 components: axis of symmetry and angular momentum vector, and then define the angular momentum part as $\Omega_p$, meaning the precession component. My question is, why? Why not orthogonal components? Why is the angular momentum component considered the precession component? Is it a mere definition?
The direction of change for the angular momentum vector follows a circle which lies in a plane defined by the torque vector and its rate of change. Calculations for that circular motion are most easily handled by using a component of the angular momentum which lies in that plane.