I applaud your research.
Torque:
Rated, working or running torque is a function of rated HP and rated speed.
Maximum torque is a function of design and is generally outside the continuous operating range of the motor. (there are exceptions)
For an Induction Motor (IM) maximum torque depends mostly on the design of the rotor, particularly the squirrel cage winding.
The resistance of the squirrel cage winding and the depth of the winding below the surface of the rotor both affect the maximum torque.
For good basic information on induction motors see the Cowern Papers.
The Cowern Papers
Look particularly at page 3; Design Letters. This shows the torque curves of different designs of induction motors.
For added research search for "double squirrel cage windings".
Synchronous types of motors;
The maximum torque of a synchronous motor may have two values.
The first limit will be the torque at which the motor drops out of synchronism.
At that point, the motor may stall or the motor may revert to the starting torque curve.
Starting of synchronous type motors;
Starting at grid frequency;
Historically, synchronous motors were started as induction motors.
The synchronous motors used a squirrel cage winding for starting as an induction motor and as a stabilizing winding when running in synchronism.
The curves on page three of the Cowern Papers show just four of the possible squirrel cage designs.
The squirrel cage winding on a synchronous motor when used for starting will be designed for short time use and may not be suitable for continuous use.
Running an induction motor at variable frequency.
What is much more important to the torque of an induction motor than the applied frequency is the slip frequency or slip speed.
The slip frequency determines the frequency seen by the squirrel cage winding which determines the torque.
I will use a 1760 RPM motor for illustration:
This motor develops rated torque when the load pulls the speed down to 1760 RPM.
That is at no load the motor runs at 1800 RPM (neglecting parasitic losses, windage and bearing friction)
As the load is increased the speed drops until at full load the motor is at 1760 RPM and rated torque.
The slip speed is 40 RPM and the slip frequency or rotor frequency is 40RPM/1800RPM x 60Hz = 1.33 Hz.
For any applied frequency, the motor will develop rated torque at a speed corresponding to applied frequency minus 1.33 Hz.
Remember that torque is an effect, not a cause.
There will be no torque until a load is applied and then the actual torque will be that demanded by the load.
I will leave a discourse on the other types of motors you mention to others.
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Ohm's law
Not just a good idea;
It's the LAW!