"Urgent" gets attached to requests for all kinds of reasons that have nothing to do with actual time pressure, someone's own anxiety, a habit of over-flagging, or simply forgetting to ask until the last minute. Taking every "urgent" at face value trains people to keep using the word loosely, and trains you to drop whatever you're doing every time someone says it.
A genuinely urgent request usually has a real, external deadline attached to something with a consequence, a client waiting, a system down, a decision that can't be made without this input. A request labeled urgent but missing that external deadline is often importance mislabeled as urgency, or someone's own poor planning becoming your emergency.
"Urgent" should mean something breaks if this doesn't happen today. If nothing breaks, it's important, which is a different word, and a different response.
A quick, honest test: ask what specifically happens if this waits until tomorrow. If the answer is concrete and bad, a client walks, a deploy is blocked, a deadline with real consequences is missed, that's genuine urgency, and it deserves to jump the queue. If the answer is vague, "it would just be better to have it sooner", that's importance, which still deserves attention, just not at the cost of dropping something else immediately.
"Happy to prioritize this, quick check so I know how to sequence it: what happens if it lands tomorrow morning instead of today?" This isn't pushback. It's information you genuinely need to triage correctly, and most people answer it honestly when asked directly and without suspicion in your tone.
This matters for you as much as for the requests coming in. If you're the one asking for something, be honest with yourself about whether it's actually urgent or just feels that way because it's fresh in your mind. Overusing "urgent" on your own requests trains the people around you to eventually discount the word when it comes from you, the exact moment you'll need it to mean something.
One more distinction worth holding onto: something can be both important and not urgent, most strategic work falls into that category, and it's exactly the work that gets crowded out if everything gets treated as equally time-sensitive. Protecting space for the important-but-not-urgent is one of the clearer signs of good judgment in how someone manages their time.