What's the difference between an airplane that stalls nose down or just mushes in a stall?
You are absolutely right that it has to do with the pitching moment (Cm) characteristics of the aircraft. Take a look at the sketch below illustrating three different pitching moment characteristics:



In all the cases, you can assume that the CLmax occurs around 12deg AOA (not presented here). Cm1 is the aircraft that would present a distinct nose-down upon reaching CLmax. This is also known as a pitch bucket, because of its shape, and is good for stall identification and recovery as the aircraft tends to un-stall itself.

Cm2 is the mushy case. With increasing aft column pressure, the AOA will continue to increase, while the airspeed may stagnate. No stall identification is naturally provided to the pilot.

Cm3 is anything from stick lightening to outright pitch up, depending on the aggressiveness of the instability. This is a common characteristic for high speed aircraft that have large LE sweep and the stall does not originate from the root of the wing.

As mentioned by Carlo, the degree of nose-down or nose-up would also be CG dependent. Aft CG would always have smaller pitch bucket and worse pitch up (if exist).