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Just my guess, but I would think your assessment of starting the clock on the subsequent snap is correct because NF: 6-5-3 advises, "Only the receiver who gives a valid signal is afforded protection. If, after a receiver signals, the catch is made by a teammate, it is not a FC but the ball becomes dead."
It is the act of catching the kick (after a valid FC signal) that causes the clock to stop.
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