You have almost got it right.
Mach3 is more or less used as the machine controller on machines that do NOT have their own controller. Mach directly drives the stepper boards.
On a machine like a Fadal you can get it with than Fanuc18i controller (which would be instead of Mach) or the Mazak machine has a Mazatrol controller (instead of needing Mach). On machines such as these your CAM software (MasterCAM) sends Gcode to the controller (eg Fanuc) and the Fanuc processes the G code and tells the motors what to do.
If you look at a machine like the Syil X3 it does NOT come with a controller that tells the motors what to do - so - you need a computer and a program (such as Mach) to do this.