Actually, it's not really that far off, and I'm not in a fightin' mood today, because I have to go crawl through a 180 degree jungle of pipes to change out a 60 series B&G that some jackass hung in there 30 years ago...
But anyway, the capacitor corrects the phase angle of single phase power, to apply more torque. In this case, it doesn't actually "create" a third phase, but allows to single phase power to "straighten" out enough to cause more torque.
And, Mark, the start winding (And start capacitor) IS deenergized when a motor reaches full load.
Put your amp clamp on the start terminal of a running motor, then put it on the run and common terminals.
No, capacitors don't "switch" anything.