Sounds good, but it will introduce new problems: 1) someone called and said "I sent you that urgent email; didn't you get it?!?" and you had forgotten about the rules you set up for email receipt. 2) Services that send email notifications now have to add another reminder so people may have to check their email rules in addition to the Spam folder in case mail is "lost" there.
The fact that someone else thinks an email is urgent does not mean that I'll think it is.
And if it really is urgent, they can call me. Unless you're the kind of person who both sends an email and then immediately calls me about it. If you're that kind of person you quickly will be informed of how much I dislike that. If you persist, I'll take advantage of my freedom as a contractor to choose my own projects, and you'll soon not be a person that has any reason to interact with me.
If it's urgent you should still call first to confirm that the person is available and able to deal with what you need them to deal with and that what you're sending them is what they need.
Anything else risks wasting a bunch of people's time going back and forth which if it's genuinely urgent (it usually isn't) everyone should want to avoid.
If someone's likely to call you to follow up on an "important" email that isn't important, turning off email doesn't help. You'll still get a vibrate/ring, look down at your phone, and decide whether or not to pick it up -- exactly the kind of distraction we're trying to avoid here.
The problem with the call-me-with-only-urgent-stuff approach is the Boy Who Cried Wolf problem: if people are constantly calling you (because they know they're not going to get a reply until later in the day, and they need it <whine>noowwwwwwwww</whine>, you'll put your phone on DND/silent. But now you can't respond to actual emergencies.
In the first case, you must take action to appease them, whether replying to the email or taking their call. (Or you could script your interaction with them; an automatic email reply or such. But that's suboptimal too). It's outside of your control.
In the second case, they aren't able to distinguish what they consider to be emergencies from what are actual emergencies; this, too, is out of your control.
I honestly don't know how to deal with these. Thoughts?
My solution is as I described. I make it clear that I am unhappy. I make it clear why. And if they refuse to listen, I arrange my work so that they have no excuse for trying to ruin my day.
So just have a manual "get mail" button that you can mash like every other mail client ever. If someone sends me an urgent email, and it's genuinely urgent, they'll follow it up with a phone call or a text or an IM, and then I can mash the "get mail" button.
