Two Limits in Thermodynamics - Reducing Driving Force and Increasing Resistance

I just wanted to clarify what is probably a very obvious idea.

Suppose I'm holding a a piston separating two containers at different pressures. If I release the piston, it will naturally move, probably quite quickly, to a new position, so the pressures in both containers are equal.

This is obviously an irreversible process.

However, people often talk of a 'reversible' process as one that occurs 'very slowly', so that each step can be reversed. But I think that's ridiculous!

I see two possible ways I could slow this process down, so that in the limit it's moving arbitrarily slowly.

In the first case, I could add gas to one container, so the pressures are arbitrarily close, and thus make the piston move arbitrarily slowly. If I kept slowly removing gas I could run the whole process, and I think this would be reversible.

On the other hand, I could keep the pressures different, but make my piston heavier and heavier - I could make it out of neutron stars! In this limit I could also make the process arbitrarily slow, but it wouldn't be reversible - I can calculate how the entropy will increase!

The analogy with temperature is obvious (I could slow heat flow by reducing $\Delta T$ or by adding more insulation), and indeed question is really this:

In thermodynamics, is it the case that reversible processes are not processes which occur arbitrarily slowly (as I've been taught and read many-a-time) but instead are processes with arbitrarily small driving forces at each stage? On a side-note, is this difference ever discussed in the literature? :)