Shouldn't this be the realm of kinetics, then, which is specified in Case 2 rather than Case 1? And how exactly does having non-0 ΔG produce a potential difference?
They are both kinetics... and both thermodynamics.
The difference between the two scenarios is what the equilibrium point is.
In scenario one, you are dealing essentially with a simple redox reaction. If you put a stick of zinc in a dilute acid, there will be a reaction because there is an electrochemical driving force for the conversion of Zn
0 to Zn
2+ and H
+ to H
2 gas. We can represent this driving force by a Gibbs energy change, which will have an associated equilibrium constant. The reaction will proceed until an equilibrium is reached and the driving force is zero due to a balance in the concentrations of reactants and products. I think you know all of this because it is simple equilibrium chemistry - the only difference is that the reaction is a redox reaction, and thus the thermodynamical quantities can be expressed in terms of simple redox potentials. (And in a way, it's really just a superficial difference - ALL reactions are redistributions of electrical charge, more or less; the only difference here is that we can make the approximation of "whole units of charge" moving from "predominantly one nucleus to another", which allows us the convenient use of redox potentials to express pretty much the same thermodynamical equations in a different form.)
In scenario two, all that changes is that there is an external potential energy applied. What this really translates into is an additional force that resists or reinforces the predominant (natural, you might say) driving force for the reaction. Essentially what it does is change the equilibrium point to favor more the products or the reactants. Remember: ΔG is a potential energy concept, which means it is therefore a force concept. The Gibbs energy tells you how a reaction will behave because all systems try to minimize potential energy. By adding an external potential energy, you are therefore adding an additional force, which will shift the reaction equilibrium accordingly - and always away from the "natural" equilibrium point.
This is essentially how you charge a battery. You apply a potential difference across a battery cell, which shifts the equilibrium because you are changing the reaction's driving force by adding a new force to the equation. But what you are really doing is storing (potential) energy in the cell, because when the external potential difference is removed, now you are suddenly at a nonequilibrium position, and there is a natural driving force to return to the original equilibrium point. So you remove the charging source (the external potential) and the electrochemical cell wants to return back to its starting point, and it'll do this by letting electrons flow back the way they came. This electron flow is harnessed to do work elsewhere.
These charge/discharge cycles cannot be repeated indefinitely, because there are usually other irreversible, minor processes that happen during each cycle (typically the quality of the electrode declines over time, because the metal doesn't re-deposit perfectly - this results in imperfect condition, the generation of excess heat, corrosion of other parts of the battery, etc.).