Ok, so maybe the motor itself won't be more efficient but the circuit is.
As for wiring diagrams you can look for a similar one or use the information provided by the PO, he's unlikely to have written it down without understanding what he was talking about. If the motor is working now it's probably wired for the service you have it connected to so I wouldn't change any wiring until you rule out other potential problems.
Often when a machine sits for a long time it will need at least a partial tear down and rebuild to make things work right.
There are many things that could effect lathe operation besides the motor itself, just because it looks clean doesn't mean everything is set-up and adjusted correctly.
How are you measuring rpm?
Have you disconnected the belt and run your motor with no load?
Have you cleaned and lubricated the spindle bearings?
Bearing preload?
Changed or at least inspected the belt and made sure it's tightened correctly?
Does the spindle get hot when running?
Does everything turn freely when the belt is disconnected?
If you're looking for a replacement motor you should be able to use anything with the correct hp and frame size. I still recommend a three phase unit with VFD for convenience of changing speeds if replacing the motor but I suspect something else is causing your problem as electric motors tend to either work or not, if the magic smoke is still in there look elsewhere for your problem.
Here is the operations manual for your machine, read it and pay close attention to any sections dealing with adjustment and lubrication.
Cheers,
John