To add another piece into the mix, when does a cut fastball become a slider?
I know some of what GBM is talking about in terms of the CB vs. the Slider. My son throws a breaking pitch that has been called many things. His PC used to call it a cut fastball, sometimes PG will call it a slider, sometimes a CB and sometimes both. He basically holds it the same way, although sometimes, it will be later breaking and sometimes more sweeping. He throws from low 3/4, so the pitch does have 2 plane movement. He also throws a CU that will drop off the table when working to the point coaches have asked it was a split finger pitch.
I think the breaking pitches (CB, Slider and even cut fastball) are all variations of the same pitch, it's just a matter of degree. It really depends on the movement that will determine what it will ultimately be called. CB will have the most break and movement, Slider will have less than that and it will occur later and the cut fastball will have the least movement and the latest break.
The problem I see with calling all of these "off-speed" pitches is that you would have to consider a CU as an offspeed pitch and I don't put them in the same category as the others. I would consider breaking balls to be the CB, Slider and maybe cutter (although I'm not real sure on that) and off speed to be the CU and maybe even the split finger (not sure about that either).
Anyway, just my opinion as a lay person who has been watching, listening about and asking about pitching since my 18 year old was 8. I sure don't know everything, that's for sure.