Based on my experience, it mostly translates to whatever task the person thinks it's important for them. Sometimes people will mark the task as "urgent" just to dump it on someone else.
The other thing is, in critical emergencies, you should work slower than usual. This means muting all notifications and taking your time. Rushing to get things done can exacerbate the issue.
Best engineering managers shield the team in those situation and simply reply with: "thanks, we've been notified and investigating".
If you’ve already read the email it’s less efficient to come back to it later
mainly i just wanted to stress that nothing in an email inbox is an "absolute emergency".
it trains people to expect things.
boundaries.
1) someone having an emergency needs to contact me specifically.
2) that someone only has my email, and no other possible way of contacting me (or someone else that can contact me).
you can shorten this to "never reply instantly", because an "absolute emergency" will never be communicated by email.