Definition of reversibility for a non-adiabatic process

I would like to find a clear definition for the following expression:

System A undergoes a reversible transformation

when the transformation is not adiabatic (when system $A$ is not thermally isolated). I’m going to try to explain what I have in mind. My question is something like:

• do you see any limit, mistake, misunderstanding or confusion in my attempt to find a definition?
• have you found a more satisfying or relevant definition?

The definition of reversibility for "the transformation of $A$ + surroundings" is ok. This is the definition of reversible for an adiabatic process. You can do it thanks to $dS=0$, statistical equilibrium at each stage or ideas such as "you can go backwards".

When heat exchange comes into play, I sense a certain confusion. We know that heat exchange from a source $B$ of different temperature is irreversible for $A + B$. But from the point view of $A$ only, does this matter?

Here is an example:
You have a heat bath at temperature $T_{bath}$ and an ideal gas (say with $N$ particles) initially at $(V_{gas}=V,T_{gas}=T_{bath})$. Then:

• You expand it adiabatically (and reversibly) to $2V$.
• let heat flow from the bath into the gas until equilibrium (at constant volume)
• compress it adiabatically (and reversibly) to $V$
• let heat flow from the gas to the bath until equilibrium (at constant volume)

From Clausius’s point of view, this is a reversible cycle:

$$oint frac{delta Q}{T_{gas}} = 0$$

Thus, you could say the process is reversible (for $A$) even though some irreversible heat exchange happens. I’m tempted to say irreversibility (for $A$) cannot be about heat exchange with the environment. As long as the work is reversible ($delta W = -PdV$), the process can be called reversible. For a reversible process, you have $delta Q = TdS$ but this is not a property of the heat exchange, only a consequence of the fact the entropy is increased by the heat only (no extra growth by some irreversible work).

Thus, reversibility would be about work only. A transformation could be said to be reversible if the work done on the system is reversible:

System A undergoes a reversible transformation if the work done on the system is predicted by the generalized reversible force: $delta W = -PdV$ with $P=-left(frac{dU}{dV}right)_S$. This is equivalent to $delta Q = TdS$.

Does this work?