Unfortunately, the manual you referenced doesn't include a schematic. Owner manuals for this class of lathe don't seem to include them, probably because the motor controllers usually are made by somebody else. They also may use a variety of controllers, depending on whose is the cheapest at the time. That way they don't have to edit the owner's manual (and the info may be considered proprietary by the OEM).
I'd suggest gaining access to the motor controller to see if you can determine who made it: and, better yet, what model it is. KB Electronics was/is a commonly-used supplier for mini lathes. If that's what you have there are a number of schematics that can be found on the web.
Older minilathes used brushed DC motors and SCR style PWM controllers. Newer ones use BLDC motors and ESC style controllers. It's easy to tell the difference, the BLDC models typically have higher-power motors and no high/low transmission gearbox.
There's nothing wrong with the older motors, the controllers use feedback to maintain RPMs under load. You just have to check the brushes on occasion to make sure they're not going to damage the motor commutator any time soon.