An icemaker, ice generator, or ice machine may refer to either a consumer device for making ice, found inside a home freezer; a stand-alone appliance for making ice, or an industrial machine for making ice on a large scale. The term "ice machine" usually refers to the stand-alone appliance.
The ice generator is the part of the ice machine that actually produces the ice. This would include the evaporator and any associated drives/controls/subframe that are directly involved with making and ejecting the ice into storage. When most people refer to an ice generator, they mean this ice-making subsystem alone, minus refrigeration.
An ice machine, however, particularly if described as 'packaged', would typically be a complete machine including refrigeration, controls, and dispenser, requiring only connection to power and water supplies.
The term icemaker is more ambiguous, with some manufacturers describing their packaged ice machine as an icemaker, while others describe their generators in this way.