[CecilOne]

I'm uncertain about how these fit together or separate.
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The "actively" is something that is not a normal move while performing his/her duties in the batter's box. Attempting to strike the pitch is part of the duties of a batter and that includes the entire swing, from start to finish.
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The batter is permitted to swing at the ball and in most cases there is a follow-through associated with that swing. Unless you observe the batter do something out of the norm with that follow through, there is no violation.
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In USA Softball, Rule 7-4-I and Rule Supplement 24 says that if a batter swings and misses the pitch, and then on his/her follow-through hits the ball, or hits the ball after it bounces off the catcher or his/her mitt, then the ball is dead and runners cannot advance.
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I guess the natural duties
of the batter theory would not apply in that case and seems that it may not, in fact, apply when the batters follow through knocks the ball out of the catchers glove when there is a steal being attempted according to your quotes of the other rule sets.
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RS #24

C. If the batter swings at and missed the pitched ball but
1. Accidently hits it on the follow through, or
2. Intentionally hits it on a second swing, or
3. Hits the ball after it bounces off the catcher or mitt/glove.
The ball it dead and all runners must return to the based occupied at the time of the pitch (FP, SP w/stealing and 16"SP). In (2) and (3), if the act is intentional with runners on base, the batter is called out for interference
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