|Robert Zimmerman, aka Bob Dylan|
The contention that poetry is defined by quality is easy to disprove. Is "The Tay Bridge Disaster" prose? Let's face it: bad poetry exists. It's not a oxymoron. In fact, it's everywhere. Whole institutions and myriad publications are dedicated to the presentation, if not the preservation, of bad poetry.
And, no, it doesn't matter which is written first, the music or the words. To wit, the music to this song was composed centuries after the words:
Meanwhile, the lyrics were not added to this old folk tune until 1971:
Nor does it matter if the same person is writing both music and lyrics, even if at different times, as was the case with "Suzanne", published as textual verse in 1964/1965 and not performed as a song until 1966 (and not by its author until 1967).
There are books and courses on Cohen's and Dylan's lyrics, but does literature have to be read? If so, movies and plays are not literature, a thing that most primitive societies could never have produced. Poetry, a thing which is (with rare exceptions) meant to be performed, could not be considered literature. Fortunately, most definitions include the specification "work or production". Thus, if Shakespeare is literature, so is Leonard Cohen, Bob Dylan, Ferron, Bruce Cockburn, Gordon Lightfoot and, yes, the St. Exuperian John Prine.
The idea that adding music somehow precludes verse from being considered literature ranks second on the list of ridiculous human notions. (Right after the 22nd Amendment, of course.)