Running the fan with the engine off (alternator off) is the least efficient way to run. The lead acid battery is extremely inefficient at charge/discharge, so running the fan from the alternator output is better.
Regardless, the fan should not be manually controlled. It needs to come on when the engine is too hot, and switch off when temperatures are ideal. The driver should not be distracted with trying to control temperature when the thermostat does an excellent job of this already.
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