I'm looking a general definition of a reversible process for a (closed) system that wouldn't need to say anything about the environment and would be based on the intuitive notion of "reversing".
For adiabatic (no heat exchange) processeses, it can be done:
"An adiabatic process between two equilibrium states consisting of a controlled change in some external variable $V$ is reversible if, when done in reverse, the system goes back to the same internal state".
I wish I could find such a simple definition even in the case of heat exchange, without mentionning changes in the environment. The definition (from here) seems to be fine:
The practical definition of a reversible process is one for which the system (no mass entering or leaving) passes through a continuous sequence of thermodynamic equilibrium states.
With this definition, you don't need to mention "at equilibrium with the environement". The question arises when some heat exchange happens and the environment is not at the same temperature. There is a temperature gradient: either the gradient is outside the system, and then the process is reversible (for the system), or the gradient is inside the system and then the system is not at equilibrium.
Still, if this definition was to be summarized by a single word (linguistically), it would be natural to call it something like "quasi-static". There is no notion of "reversing". I wonder if it is possible to find a definition based on the intuitive notion of "reversing" without mentionning changes in the environment.
(Statistical physics is welcome in answers if required)