In general, the distinctions between overdrive and distortion tend to fall into these criteria (IMO):
1- Overdrive is meant to compliment and enhance a tone that is already in or near breakup.
2 - Overdrive retains some element of clean signal layered into the clipped waveform.
3 - Distortion doesn't require 1 or 2.
The OCD is an overdrive by that definition. However, I would consider a Rat more of a distortion even though it has some elements of 1 and 2 and is pretty similar to an OCD. Both have non-inverted gain stages which means some clean is layered in the result. But, they sound radically different.
The biggest difference (again, IMO) between soft and hard clipping is that hard clipping can be more responsive to picking dynamics. For example, in the OCD you can actually get a fairly clean sounding signal if you pick very lightly. As you increase the picking dynamic the output clips proportionately and the overall perceived compression increases. This can also be true for overdrives, but sometimes to a lesser extent.
Then again, I don't think these are very important or needed distinctions.