Is a discharged battery in a state of chemical equilibrium?
Post-script: After recently encountering a related post I re-read my answer here and realized I could have provided a much clearer and more useful definition of equilibrium (rather than my somewhat philosophical earlier answer) that encompasses the different cases you present: in thermodynamics, a system is in an equilibrium state when no lower free energy states are accessible. 

Therefore, the two beakers, just like the exhausted battery, are at equilibrium, because there are no accessible lower energy states under the defined conditions.



You can ask yourself: if the universe is expanding (that is, if it is in a dynamic state), is thermodynamics truly ever applicable? The answer is that in science we need to take approximations, during measurement or when extracting information from data. So it is in thermodynamics, which works well enough even though on long enough scales the approximations fail (on timescales of the "in the long run we are all dead" sort). 

This is of course a tricky point. We often need to distinguish between "real equilibrium states" and "kinetically trapped or metastable states". However it is useful to invoke a "duck-typing" rule applicable to thermodynamics problems: if the measured properties of the system remain constant and homogeneous during the interval of the experiment, then the system can be considered to be at equilibrium during that interval. This definition is particularly valuable since thermodynamics is an empirical science. Caveat: that doesn't mean you can be sure that the system is in the state you meant it to be in (!), since if you wait long enough you might end up in another state (which might be the one you actually wanted the system to be in!). Perhaps more important is reproducibility: can you put the system into the desired state, reproducibly?

To summarize: the concept of equilibrium, like that of reversibility, is subject to the practical condition that no variations be observed across the system or over time during the course of the experiment, when the system is in the proposed state. 

As regards a battery, if you don't see a change in the potential, you can regard the system as being "at equilibrium", at least temporarily! Again, the definition of equilibrium is not as strict as we'd like to think. The materials in the battery, left exposed to the environment long enough, will rust and fall apart, leading eventually to  other "equilibrium" states.