The language doesn't just cover serious criminal harassment or intimidation, though. It covers anything designed to annoy, discomfit or offend an audience that could range from the next-door neighbors to nerf herders from the outermost frontiers of Siberia.
Stalkers making a phone call or sending a text message have a pretty good idea who is going to receive the message.
Stifling open debate, not just cyberbullying
Online there's no way of knowing who the audience is.
Online the act of purposely annoying people is known as "trolling," as in being a troll, fishing for trouble. A troll is a starter of arguments, hurler of epithets, expounder of racist or debunked theories or thrower of monkey wrenches into the machine.
In French it's pronounced "provocateur."
In English (which Arizona requires) the historical and literary term for trolls has varied: satirist, iconoclast, crank, critic, whistleblower, protester, heretic, individualist, maverick, radical, lunatic and "my learned friends in Congress."
For most, the correct term is a##hole, but the language of the bill doesn't give any way to determine which annoying iconoclast in an online forum is a provocateur and which is an asshat.
There's also no way to draw a line between the words "terrify, intimidate, threaten, harass" – which are legitimate crimes defined by courts and laws elsewhere – and the words "annoy or offend," which are what happens around the dinner table at family gatherings or when anyone mistakenly brings up politics during coffee hour after church.
Banning anything likely to annoy anyone who might read something posted online would eliminate most open discussion of any issue, political, religious or social.
It might keep some from being offended, but it would keep far more from being enlightened by the revelation that not everyone believes the same unquestioned opinions they do.
Annoying others is the point, not the offense
That opportunity for discussion – and the chance to be annoyed and offended by it – is the reason our whole political system was set up as a system of checks and balances. It's a way to let opposing extremes come together and compromise on reasonable, practical decisions they both hate, but that benefit the country as a whole.
Most trolls are not doing anything noble, or literary or even defensible by pissing people off on purpose. They're just trying to stir up a little excitement and prove their significance.
Some are legitimate freethinking, counterintuitive interpreters of culture keeping up the American tradition of taking no idea as too sacred to be attacked, even if only to see if it makes enough sense that it can be defended.
Even some of them are asshats.