I don't think "composition" is correct in this context.
To compose = to prepare a text for printing by setting up the characters in order;
To write or phrase (a letter or piece of writing) with care and thought:the first sentence is so hard to compose
"The word 'composition' may mean different things in different circumstances. It could be a personal narrative, a short work of fiction or prose, an essay, a dramatic work or a poem. Each of these works has its own set of rules."
Yes there's a difference. An essay normally has a specific subject, like a history essay - Write a 250 word essay on WW2.
Whereas a composition normally implies more creativity and freedom, less specificity. Like - Write a 250 composition on your ambitions.
The guidelines for compositions are a lot less strict.
If in doubt call it a piece of writing. That fits every scenario, though is slightly less idiomatic at times.
I agree with ydnas' distinctions, but in practice in British schools and universities the terms are used synonymously for students' work. Redaction, with an acute accent added over the E, should be mainly left to the French who use it frequently, but if used in English on rare occasions it implies an improvement or revision of what has been written.