A stepper motor is generally used in open loop control systems and is driven with pulses. By carefully counting the steps (square wave pulses) going from the output board into the motor the angular movement of the motor shaft can be carefully controlled. A servo motor is used in a closed loop control system and is a DC motor with a rotary resolver and a tachometer attached to the shaft. When electircal power is applied to the motor it simply turns. The resolver and tachometer send information back to the computer so it knows when to stop. Both motors are used for precise positioning. Most steppers are generally low horsepower but the servo motors can be powerful, massive and expensive. All you need to drive the spindle on your lathe is a regular, ordinary motor. If it turns it will do the job perfectly. A DC motor like your Hamilton Beach, can be controlled with a common sewing machine pedal or an inexpensive (Harbor Freight) router controler. I personally prefer small AC induction motors and control the speed with pulleys and belts. I like them because they are quiet.