I think "No Catch" is more clear. What's the merit behind "Gotta get him!"?
"Gotta get him!" Means exactly that. The catcher either needs to tag him, or throw him out.
The main thing is to give the visual mechanic so everyone can see. If the batter or catcher are clueless with your mechanic, or unaware of the count, that's their issue.
Similar to the infield fly (mechanic and verbal for the runners), giving a verbal on the dropped third strike is providing a curtisey to the defensive team of what needs to be done to retire the BR.IMO That phrase can be interpreted as coaching F2 and is not what most players are expecting to hear. Inventing your own mechanic usually backfires.
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I understand what you mean and I respect it. However, keeping the integrity of the game is why we do what we do. I personally don't believe that any player at any level would interpret such phrase as coaching.
But as stated in a previous post, the visual mechanic is the most important thing to do in this circumstance.