Using the same signal for both timers is fine. I drew it out that way at first, but the drawing was more confusing, so I redrew it with pony motor related on the right, and the rest on the left.
My reference to the 2014 NEC is for when the caps "are not an integral part of the rotary-phase conversion system, but are installed for a motor load..." This applies if the caps are located at the machine, not when they are mounted in the RPC system enclosure.
My system is a capacitor start system, not a pony motor start, so yes, it is different.
In the Larson drawings contactor 1 controls power to the pony motor. Contactors 2,3, and 4 control power to the idler, and also connects the capacitors to the idler. Those capacitors go between L2 and L3. Contactors 5 and 6 connect L1,2 and 3 to the load terminals of the RPC. The second page shows two timers, with relay 1 controlling the pony motor through contactor 1, the idler through contactors 2,3 and 4. Relay 1 also powers relay 2, which controls contactors 5 and 6. Both timers are time delay relays, but we do not know what times they are set for.
NEC 2014 455.21 states that power to equipment shall not be supplied until the RPC has been started. Timer 2 and contactors 5 and 6 provide a delay, so power "out" is not supplied until idler is spun up and powered. This insures that single phase power cannot go to equipment if someone happens to start a machine at the same time the converter is being started. My system does not have this provision. Like your situation, there is a time delay while I go to a machine so I am comfortable with this.
The capacitors are fine with the full power, assuming they are of the proper voltage, and the wires are sized for the current they carry.
The most practical way to simplify your system is to ditch the 200 amp two pole contactor and replace it with a 3 pole contactor. You can then wire in the caps between the circuit breaker and the 3 pole contactor. In doing this you only need the one timer you are already using. See the drawing below.
I just went back and looked at an earlier post of yours in which you mention having a 3 pole contactor, and you may have been suggesting what I drew out above. My apologies for not picking up on that.