An exercise book in the UK is simply any blank notebook for the student to write in. It may be plain, ruled or squared/graphed. I've never heard of the term 'composition book', ever. Even the ones for music class with pre-printed staves were still called exercise books… even though we were learning composition.
They would be most commonly be simply stapled in the middle, like a magazine, but the type of binding doesn't really dictate the name. Generally they're all exercise books, for doing exercises [work] in.
Older students would often transition to loose-leaf ring-binders.
[This from when Noah was a boy; I presume everybody has an iPad or laptop these days;)
Strangely, as soon as you leave school you never use the term exercise book ever again. It seems very school-specific. It's a term I hadn't even thought about in decades, until I saw your question.