Western concert music, with its centuries-old tradition(s) of scholarship and study, has some very specific terms for very specific things: Essential Structural Closure, a modulation to the flat submediant, and my go-to bugaboo hexachordal combinatoriality — these all refer to pretty specific things and those specific things only.
Western concert music, being a rowdy and living tradition practiced by untold numbers of people with wildly different levels of formal education, also has some terms that can pretty much mean anything you want.
"Arranging" falls decidedly into the latter category.
I think most people, even non-musicians, have a pretty firm grasp on what it means to compose something. The specific process will vary from person to person, but at its heart, you go into a room with a blank piece of paper, and you come out with a piece of music. For some people, that involves literally sitting at a piano and plugging away, for others it involves fiddling with audio samples in a computer program, but either way, the result is much the same: The composer is the person who decides what sounds happen when*.
Arranging is much more nebulous.
At one end of its possible meanings, you have what can also be called transcription: Taking a work originally scored for one ensemble and scoring it for a different one, as with the concert band version of Leonard Bernstein's Overture to Candide compared to the orchestral original. Calling something a transcription implies a high degree of fidelity to the original source material. The instruments change, but the form remains the same. If you run across something labeled as a transcription, you wouldn't expect to find different harmonies or new countermelodies, at least not in the way that most people use the term today.
At the other end of the scale, you have things like Jimmy Mundy's arrangement of Louis Prima's "Sing, Sing, Sing" for Benny Goodman, which alters almost every aspect of the original**, up to and including the melody itself. (Few arrangements take this degree of latitude in the concert music world, what with our general reverence for scores and source material, but it's absolutely common practice in jazz and other less score-based musics.)
A lot of things fall somewhere in the middle. When William Schuman arranged his New England Triptych from orchestra to wind ensemble, he transcribed the first two movements with minimal alteration, but completely revised the third, nearly doubling it in length from its original incarnation. Gustav Mahler's piano parts for the piano-vocal versions of his Des Knaben Wunderhorn songs are not simply reductions of the full orchestral score from his orchestral arrangements.
Then, too, there are cases where the orchestration — the determination of which instruments play which notes — is so striking, so specific that, even tho the melodies and harmonies of the original are preserved in every detail, it seems misleading to call the new work "just" an arrangement of the previous one. The most iconic instance of this is probably Anton Webern's arrangement of one of the fugues from Johann Sebastian Bach's Musical Offering — Webern's meticulously detailed and relentlessly modern scoring transform Bach's exacting counterpoint into a psychedelic whirlwind of motives and fragments; it's an arrangement, sure, but one so strongly marked by the arranger's aesthetic that one almost wants to give them co-compositional credit for the resulting masterpiece. (You don't need two different people, either. The end of Igor Stravinsky's Svadebka (aka Les noces aka The Wedding) is another place where "the notes and rhythms being played" and "the instruments playing them" seem impossible to separate; orchestration and composition become one. (The passage in question, where frenetic energy gives way to the purified ringing of bells, begins around 8'17", but the whole thing is worth a listen if you have the time.))
Given this nebulosity, it might be tempting to dismiss "arranging" as a useless term. To the contrary, it's precisely this nebulosity that makes it useful. I can say that I'm "arranging" some of the music from Window Full of Moths for wind ensemble, and I don't have to clarify that, while the second movement is essentially a transcription of "Beside You", the third is a mixture of the opening of "Hey" and a radically altered version of "Stop Dreaming", with some new material thrown in to stitch them together, and then —
Arranging encompasses all these things in a way that gets at the core of what I'm doing: This music existed before. I'm changing it in some ways so that it exists in another form as well. Some of these changes are small, and some of them are big, but for the most part, we neither need nor want to get bogged down in the nitty-gritty specifics of which are which. Having a catch-all like "arranging" lets us get away from worrying about arbitrary distinctions on a continuous continuum and gives us more time to spend actually making art.
*There are, of course, exceptions where composers leave room for other people or things to make some of these decisions. For now, I'm simply ignoring this type of composition, but I think it can be subsumed into this framework with little difficulty.
**I am not actually 100% certain that this is the original arrangement given the sheer number of recordings of this tune and Spotify's spotty bibliographic information. If someone could confirm this or provide a link to the actual original arrangement, I would greatly appreciate it.