Equilibrium potential means: all reaction went to (possible*) completion, all electrodes are charged and in equilibrium with their redox systems.
*if you short circuit the cell, reaction can proceed further, but we don't allow it.
Fast kinetics in this context is connected with the fact when you disturb the cell (for example by shorting it for a limited amount of time)), it has to reach a new equilibrium state. If the system is fast, this equilibrium can be reached quickly, if it is not, we have to wait before being able to take the new measurement. Otherwise we will see a drifting potential, which will converge to the new equilibrium value.