You either need to narrow the definition or carve out explicit exceptions. If you are carving out exceptions, the definition is just “people we don’t like”.

I guess that is pretty close to the definition prosecutors like to use in the US.

That they're similar in this manner doesn't mean that we can't have laws and law enforcement because they're inherently violent and tyrannical or anything, but it does inform how seriously and dispassionately we should approach their means and ends. I don't think that it's particularly radical to call an abuse of judicial power terrorism. What else to call, for example, a judge who robs entire communities of their youth in exchange for backroom deals with private prisons?

You have no evidence that most school shooters communicated such a goal (although there might be an outlier). Or if you do, I'm curious what it is.

Terrorists have a tangible goal, typically coupled with demands, not some hidden intention you make up to shoehorn a person into the definition. Even Elliot Rodger, who had a "manifesto", did not communicate any sort of demands or goals.