the ram's-horn trumpet: a curved animal horn sounded to give a loud, carrying signal
The shofar is not a melody instrument but a signal instrument: a hollowed ram's horn (or wild-goat horn) that makes one piercing, far-reaching blast. Because it could be heard across a valley or a city, it became Israel's way of getting a crowd to act all at once, and almost every use of the word turns on that single function of summoning or alerting people.
What changes from place to place is the occasion, not the object. The same horn rallies an army or warns a city of attack, opens a festival or the Jubilee year, crowns a new king, and marks the moment God draws near. These are best understood as different settings for one instrument rather than different things the word means, which is why they share a single core.
The one feature that ties them together is urgency. A shofar blast says "now, everyone, together," whether the thing happening is a battle, a coronation, a sacred assembly, or the presence of God on the mountain. Translators reach for several English words (trumpet, horn, cornet) precisely because no single English term keeps both the rough animal-horn sound and the alarm-like force.