Gary,
In your irst post you show 3 diagrams. Take the one with the switch S1.
This arrangement without the switch and with the capacitor permanently wired, will give you perfectly satisfactory operation. You have to be careful to select the value of the capacitor to balance the 3 separate phase voltages to be the same. For the star connected motor, your phase voltages will be 138 volt. Thus the design voltage of the motor would be 138 volt for each phase winding.
The size of the capacitor would need to be large enough to carry the required current. A typical figure for capacitance is around 70 microfarad per horse power.
I have a siemens 6 HP 440 volt DC variable speed motor with an external cooling fan. The fan motor is 3 phase of about 30 watt. The series capacitor I use is about 2.9 microfarad and gives almost identical phase voltages. The capacitor I have is a paper block capacitor and handles the fan current easily. I was lucky to have a few of these 3 uF caps and selected one with the right capacitance to balance the phase voltages.
Functionally, the capacitor gives a leading current into the particular phase. The power factor of the motor windings gives roughly a 60 lagging phase and the leading phase is set to give a 60 degree leading phase. The sum of these phase currents gives a line current of zero phase. The perfection of the system depends on how close to 120 degree the phase angle can be set between the 2 input phases. For my small fan motor, the lagging phase is almost exactly 60 degree.
Hope this helps.