I'll add one thought that isn't already represented here. You added the Cambridge definition of "arrangement" that emphasizes that a piece has "had changes made to it," and ask:
If the piece of music can be played in a different way, isn't that also considered an original piece?
Simple answer: nope. But wait: can you make enough changes that it is no longer the same piece? Of course. Change all the notes and it's now a different piece. So how much change is necessary for it to count as an original piece?
That's a question that can't get a hard and quantitative answer, but the distinction exists. Vivaldi's Four Seasons has been arranged for probably every instrument by now, and most of those count as "arrangements." But Max Richter deconstructed the elements of Vivaldi's work and recombined them in the work Recomposed by Max Richter: Vivaldi – The Four Seasons. Although he uses Vivaldi's material, and one might not call the resulting work an "original composition," it is unarguably a new and distinct work.
Wendy Carlos, in Switched-On Bach, recorded J.S. Bach works on Moog synthesizers. (Disclaimer: I'm not intimately familiar with the album, and copyright issues make it difficult to access. My perception is that she didn't significantly add to Bach's works, e.g. adding drum beats etc., simply arranged them.) While many shocked purists would deny that the result is "the same thing," this seems like a clear case of arrangement. (A similar experiment is "Cans and Brahms," in which Rick Wakeman straightforwardly crams the third movement of Brahms' fourth symphony into three keyboards (or is it four?), changing few if any notes in the process. The impact is certainly different than a symphony orchestra, but it's hardly an "original composition."
What about remixing? When a band writes a head-banging hit and then, 20 years later, releases a gentler easy-listening cover so their aging fan base can shop to it, is that an arrangement? (Probably.) If their song is chopped up and glued back together in a techno remix, is it an arrangement? (Arguably not.) Is it an original composition? (Uh... arrrguably not? It's a derivative work.) What about mashups? When Danger Mouse combines the Beatles' White Album and Jay-Z's Black Album to create The Grey Album, is that a new work? Well, all sorts of legal teams got excited about the question, but from an aesthetic standpoint, I think the combination is clearly a different "thing" than the two component "things." It's certainly not an "arrangement"!