What should be the phase difference in the two windings of a split-phase induction motor?
Single-phase motor designs are a difficult compromise. They are designed in various ways depending on which aspects of performance are most important for a given application. See my answer to your question: Why 2 pole induction motors don&#39;t use auxiliary windings while running (while others do)?
All single-phase motors are subject to vibration. The stator magnetic field pulsates rather than rotates as it does in a three-phase motor. Mathematically, the pulsating field can be fewer as two magnetic fields that rotate in opposite directions. Once the motor starts to rotate, the resulting torque increases in the direction of rotation. Torque pulsation and vibration can be reduced by distributing the windings as well as possible to make the shape of the field more sinusoidal as mentioned in an answer to your question: What are the properties/advantages of concentric winding?
The phase difference between the main winding current and the auxiliary winding current can approach 90 degrees with some capacitor value selection, but only for a particular load torque and point of the torque capability vs. speed curve. The capacitor value, the winding design and the starting scheme must be selected according to the best performance for the application as mentioned in my answer to the first question linked above.
Single-phase motors are avoided by industrial users who have access to three-phase power. One of the pioneers of AC motor design said something to the effect that the best single-phase motor design is not as good as a below-average three-phase motor design.
The best single-phase motors are designed for one spee, one voltage and known load characteristics. A motor such as yours with multiple windings that can be reconnected in various ways has forfeited much opportunity to design for good torque vs. speed performance, high efficiency and low vibration.