D3K Explained To A Coach - Letter One

[whiskers_ump]

NFHS game: Situation 1 on [1B] 2 out, 3-2 count.
Pitch bounces in front of HP and is cleanly caught by the catcher on the
bounce after the batter swung and missed. Defense leaves the field while
the batter is standing staring at the dugout unsure of what to do.
After teams exchange sides, offensive coach of previous inning goes to
PU and asked - "was the batter entitled to attempt to take first on the D3K?"
PU replies, "No, the catcher had cleanly fielded the pitch after the pitch
became the 3rd strike after the batter swung and missed." He further explained
that the pitch was not a strike until after the batter swung at the ball,
and that the catcher had then made a clean catch after the ball had become the
3rd strike.
Coach was not buying this, has sent a letter to the chapter that I now
have to answer. [No, I was not even there] [Rule Interp for chapter]