Again, without knowing the details of your summer league rules, I would have to agree with the actions of the official. Throughout the game you have a running clock, including FTs. Your exceptions state the clock will stop for T.O.s and start again as it normally would after a dead ball. You are inferring that you believe that should be a throwin (except you added the line regarding rebounds on final FT shot. If that is the case then you agree that it does not require a throwin to start the clock "normally". Therefore I agree with the official to start the clock when the ball becomes live, which is when it is given to the shooter; until you reach the 2:00 minute mark in the half.
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