The drawing was quite well done. Your "chicken scratching" would have been fine. As a rotary converter, it is one as is. The only difference between what you have and a true rotary is the capacitor(s). And the switches. . . The capacitors would be split, one for "run", the other as a "kicker" or starting cap. The absolute values will vary from motor to motor so I can't advise what value to use.
Unless you need variable speed, I would not advise a VFD. What you have will work fine, as is. Just replacing the light switches with motor rated relays and a decent enclosure are my only concerns. A "3 wire start" would be a nice addition. And a relay for the kicker should you opt to add one. And, very important, decent insulation on the motor leads. But truth be known, I think you had already seen that. . .
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