This probably is stuff for the Tech Log, anyway:
the centrifugal force, which I always have been learned doesn't really exist. But according to the FAA [...] that is the force which is the opposite of the centripel force. In my opinion, that can't be correct.
It is correct in that, if you look at the vectors, they are equal and opposite. The point is that the centrifugal force is fictitious, i.e. it's not really exerted by any body; it is the force that
IF it were applied to the aircraft by some means, it would balance all the others, allowing the aircraft to achieve equilibrium, i.e. moving straight and level. Given that nothing really exerts on the aircraft a force like the centrifugal force, the real forces are not in balance, which is why the aircraft turns in circle.
So, if centrifugal force does not exist, what is causing the ball in the inclinometer to go left or right?
Its inertia.