I tend to think of eddy currents as a circular current that forms in a conducter (eg. a plate of copper) to oppose change in magnetic flux. Both of those generators have an induced current according to lenz's law but I believe an eddy current is more specifically one formed in a conducting surface rather than in a wire.
eddy currents are produced in the iron core of the rotor whish is concentrating the magnetic field. They usually split this core up into small insulated segments, to reduce the size of the eddy current