This is how it works, the motor has two windings, a run and what's called a start. The run has heavier wire than the start so the start has more resistance. When power is applied to the windings the higher resistance of the start will shift the phase angle just a little starting the magnetic rotation in the stator that acks on the iron core of the rotor. When the rotor starts turning it builds inductive reactance or induced resistance, the induced resistance drops the current the motor needs.
To increase the starting touque a run cap is added to the start winding shifting the phrase a little more increasing the motors touque. The run cap is designed take the current of the start winding, when a start cap is added to the system it shifts the phase even more for more starting touque but it can't take the current for very long so so it needs a relay to take it out of the circuit after start up.
The start cap doesn't reduce the inrush current because all motors are close to a dead short at startup but it makes the high current draw shorter because the higher touque makes the motor start faster.
The manufacturer of the compressors designs the motors and decides what run and start caps and relays it needs for best operation not aftermarket suppliers so if you need to replace or add a start cap or any cap use what is recommended by the manufacturer, the timing of the start cap is very critical, staying in the circuit to long and the current draw goes up not long enough and it takes longer for the motor to come up to speed so higher current draw. Also if the start cap is in the circuit to long and it will go up in smoke.
Denny