Does lowering the regulator setting of fan minimize electricity usage
In general, yes slower speeds use less power.  The fan motors typically are synchronous AC motors and use different windings or capacitors on the motor to create different speeds.  There isn't a "dimmer" circuit or resistor speed control when you use the pull chain switch.  

If you try to use a dimmer switch instead to change the speed that isn't designed for fans the quality of the power (power factor) can drop which can lead to additional inefficiencies.