We need a word or phrase to refer to intentionally (and usually only slightly) misquoting something for the sake of a perceived æsthetic improvement in the context in which the quoter wants to use the material. Okay, we do have the word paraphrase, which is close ("To paraphrase the great fooer Bar Quux ..."), but I don't think that's quite it—the Wiktionary definition mentions "often to clarify meaning" and I'm definitely talking more about petty differences in word choice than actually clarifying meaning.

There's this wonderful quote by Eugene Gendlin that I often remember as "People can stand what is true, because they are already doing so," but it turns out that Gendlin's actual phrasing was "for they are already enduring it." If I want to use my slightly different choice of words because I think it flows better in the context in which I'm repeating it, I think that can be legitimate. (Although needless to say, in formal settings, you definitely only want quotation marks around the words someone actually said.)

Or if my hypothetical pop punk band (which would hypothetically be called either "Bullet Candy" or "Zack M. Davis and the Duty-Bound Empiricists") were to cover Taylor Swift's "The Story of Us", I would sing part of this one line in the chorus as "the twist of fate by which it all broke down" rather than the original "the twist of fate when [emphasis mine] it all broke down," not to suggest Swift was wrong to sing when, but rather because that kind of use of the phrase by which is exactly the sort of rhetorical flourish that my hypothetical band would hypothetically be known for.