What you see in the video, a diskman with the CD in it spinning, is a case of torque-free motion. The wobble in that case is called 'torque-free precession'.
(Not a well chosen name that one, because it's not analogous to what is called 'torque-driven precession'.)
In that video it's a CD in a diskman, so the mass of the diskman is affecting the motion too. But if you would release the CD to free motion, while it's spinning, then it can wobble too. That is not due to uneven weight distribution; the fact that the CD is cilindrically symmetrical doesn't necessarily mean it will spin around its axis of symmetry when released to free spinning motion. The disk can also have a component of spin around an axis that doesn't coincide with the axis of symmetry. What you then see is a wobbling. In the absence of air friction such a wobbling state of spinning motion will persist.
A spinning gyroscope on earth that is precessing is a case of torque driven precession. (As I said; not analogous to the torque-free case.)
As you observe, the usual motion pattern is precession and on top of that a smaller oscillation. That smaller oscillation is called 'nutation'.
Usually, when you release a gyroscope you try to do that gingerly, so as not to add nutation. You prefer to see perfect precession.
The thing that causes/allows nutation is not an assymmetry in the gyroscope. An ideal gyroscope can move with nutation too. Just give a spinning/precessing gyroscope any whack and you add a nutation to the precessing motion.
Generally, the nutation will dampen faster than the precession, so generally you will see the nutation fade out while the precession/spinning is still strong. (And if the spinning motion is very fast the nutation is very fast too, and the amplitude very small, so small that you don't see it with the naked eye.)
The most comprehensive discussion that I know of, of free spinning motion and the motion of gyroscopes, and how the two relate to each other, is the one by Eugene Butikov and his team:
Forced precession of a gyroscope
(The illustrating animations require the Java browser plugin)