As you know, when the catcher throws the ball back to the pitcher after a pitch no one in their right mind would say “the ball is in play”. You are confusing the concept of “in play”, to an absurd degree, with the concept of a “live ball”.
As far as name calling goes.... the shoe seems to fit pretty well doesn’t it?
And when that catcher makes an errant throw back to the pitcher allowing a runner on 3rd to score, I guess you'll continue to say that the ball was never in play.
To limit the "ball in play" condition to simply the act of a hitter hitting the ball is pigeon holing the entire concept..
There are countless scenarios where the ball is in play, runs are scored and outs recorded with a ball not having been hit......
but it always starts with the ball in the pitchers hand.