The primary area this comes up for me is PMs/texts and calls (It’s easier mentally to ignore a popup from youtube or twitter than a message from someone I know). Along the lines of what you’re suggesting, a simple solution would be to allow message senders/voice callers to *optionally* mark the communication as “urgent” or “casual.” By default, calls and messages would still come with neither flag, implying that they’re somewhere in the middle: “please respond at your earliest convenience.”
On second thought, I think it would be better to make “casual” the default. This would leave the burden on the caller/sender to affirmatively mark the communication as “urgent” or “response requested.”
Lots of ways to do it. On the receiving end, the app — or ideally, the entire platform — could organize the notifications into several levels of urgency. One could set their status at various levels of “Do Not Disturb.” ie, instead of simply muting all calls and messages, one could mute all except those designated as “urgent.” At other times, we could set it to mute only “casual” level messages and let everything else come through.