It may be good to have a circuit diagram of the ac motor to see what is the equivalent of a capacitor to split the phase. Logically there should not be any difficulty for speed control with a thyristor based regulator so long as the voltage does not go below a particular level.
I don't think anything will go wrong. However, why not use an autotransformer to slowly increase the voltage if you are concerned. I have experimented with many things, so long as theory is supportive.
Generally the ceiling fan regulator is rated for 60-80 Watts. But your AC machine rating is 370 Watts. If you connect fan regulator to your AC machine, regulator can not handle this much power and it will burn. Be careful in doing this experiment.
If you want to apply ceiling fan regulator for control of single phase AC motor then you have to check their current not power. But if they both designed for ~230V then the ratio of its power roughly reflects the ratio of its current. It means that if the power of ceiling fan is less than the power of AC motor then you don't have to adjust regulator for voltages closed to 230V because the AC stator current will exceed the rated current of regulator. Generally, the less voltage on the stator winding of AC, the less current (it concerns, first of all, the no-load mode (it is not clear if your AC motor has any load) because the value of AC motor current under load depends mostly on value of the load torque). But, indeed, efficiency of motor sharply reduces under non-rated values of voltage. It means that a considerable decreasing of voltage causes a less decreasing of current. To sum up, I would say that you can control the motor by your regulator but only in the range of voltages that much less than 230V.
To overcome any confusions and doubts (because there are a number different opinions), you can apply AC ampere-meter. Firstly, connect it series to your fan and find out how much current is. Then, connect the ampere-meter series to AC motor lined to grid (I mean connect AC motor with ampere-meter to the grid without any regulator) and learn how much current is in this case. As you understand, if the motor current is less than the fan one then you can apply regulator for control of AC motor. If it is not, you can try, nevertheless, to control your motor by regulator checking the Amperes simultaneously. I guess that regulator has enough reserve to control the motor under low values of voltages.
Is the position of "5" highest in a range of regulator? It can be so that the low positions of your regulator don't provide the voltage that need to start. The torque of motor depends on square of voltage, it means that if you reduce the voltage two times (from 203 to 115V) then the maximum torque will reduce four times at once! If there were any load or some obstacle for free moving, it could be enough to slow or stop the shaft of motor.
The picture attached with the query gives me an impression that its a old resistive control which is obviously for a load with a rating in the range of 100W to 180W.(Fan regulators are designed generally for 140W and 160W fans). So if you want to operate a 0.5hp motor then I don't think it would withstand the load. But if it withstands, then u can have 4th and 5th speed range only.
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