I am speaking as a trumpeter who started as a bugler, but did once have a turn at playing what we called a trumpet-bugle (simply because it did not have valves, and you know that the valves add extra lengths of tubing when you press the pistons down singly or in combinations, turning it into an instrument that can produce all the notes between C G C E G, and above and below those basics).
Peter Daniels has given us succinctly the correct technical distinction:
"A bugle has a conical bore, a trumpet a cylindrical bore."
"There are keyed bugles, and valveless trumpets."
The point is: the bugle belongs to the horn family, not the trumpet family (trumpets and trombones).
I won't say that a person who gives the word "bore" in their definition without defining it (not in the Oxon lexicon, except with reference to guns and engines and calibre) is worthy of another usage of that word (or rather a homophone) which Oxford recognizes; but having had this answer given to me politely for fifty years, with the expectation that I know precisely what the difference is, I suppose it is about time I could distinguish cylindrical and conical bores (and it is not the shape of their hat which does it). (-;
I see the Oxford lexicon gives *bugle-horn as a synonym; and *clarion is another word, but having a narrow tube and a warlike shrill tone it must be that "trumpet-bugle" I mentioned (which is really a valveless trumpet).
What have we got in the Bible?
Daniel 3:5, at the court of King Nabu-kudurrru-us.ur (pronounced Caractacus, or Nebukadnessar),
Aramaic QARN 'horn' or 'cornet' (King James version)
Joshua 6:5-6, at the battle of Jericho, Hebrew QEREN 'horn' or trumpet' (KJV) together with SHOFAR, 'ram's horn)
It seems that here the words refer to the same object, the horn of a ram used like a conch shell or Siegfried's bovine (I presume) horn for getting attention.
H.aS.oS.eRa (passim): reed, tube, trumpet.
This sounds nicely onomatopoeic to me: KH introduces the breath; TS, TS the tonguing behind the teeth and the spittle that accompanies the breath (a good argument for TS as the ancient pronunciation, it occurs to me right now!); and the tongued or trilled R is part of the mix that goes into the mouthpiece and is amplified in the metal tubing.
Now, in brass bands and orchestras two similar instruments are the *cornet and the trumpet.
The cornet is squatter but they both have the same length of metal, wound around so that it is not disturbing the player in front of you (prodding his back or blasting right in his ear). The cornet is sweet-singing the trumpet is brassy brash.
And whether it is a conical horn or a cylindrical/tubular trumpet it has a bell (they all open out like a cornucopia).
So the ancient Israelite metal instruments were 'trumpets', and the others were animal 'horns'. The word 'cornet' does not apply, nor 'bugle', I would think.
Recently, after an early music concert I was allowed to hold a bent wooden (!) 'trumpet' (as used by Monteverdi and Gabrieli).
Still, someone could clarify conical and cylindrical for us. Does it mean that the bugle and the cornet and the French horn are widening their hole all the way to the bell, while the trumpet keeps the same width most of the distance?
Some time ago, Helen and I, and our grand-daughters Olivia and Julia, with their mother Laurel Colless of Virginia Tech, and their father Pekka Lintu the Finnish ambassador to Washington, marched round our house each with a percussion instrument, while I used a trumpet as a valve-trumpet to play the Grand March from Aida, but also playing bugle calls, which did not need any fingerwork, only tongue and lips and spit. Then they went back to D.C. for the Presidential Inauguration, and the Ball. Did you see them there? They have now moved to the Finnish embassy in Athens, to ask the Greeks why they can not live within their means as the Finns do.