The reality of the situation, is it is "my strike zone". I am the human element that is calling balls and strikes that day.
I will admit that there are umps who will give you the image that they are arrogant when making reference to "their strike zone". But much of that has to be taken in context with the game situation.
If there is a pitch, that I think is a ball, and the coach/catcher think is a strike, or visa versa, I think the pitch is a strike and the coach/batter thinks it is a ball, a discussion regarding "my strike zone" could happen. Depending on how that conversation plays out will determine if the ump is arrogant about "his strike zone".
And remember, this is all in context with the "borderline pitch". Guess what. It is what I call it. It's "my strike zone"
One of the best complementss to an ump, is "He's been calling it there all day!" Whether you agree or disagree with the call, that is not up for discussion. Why? Because it is "my strike zone."