Uses of the word "lyrics"
I'm going to attempt to answer this question specifically in the context of modern American colloquial usage of lyric and lyrics, specifically in the sense of words sung in a song, since that appears to be OP's focus. I'm basing this on personal experience on (i.e., countless hours of discussing music with friends).
I'd define the common usage of the word lyric (singular) as:
a cohesive group of the words in a song, esp. a phrase, line, couplet or verse
Typically, a lyric is a standalone unit containing words of a song - a piece of the whole. Often the piece is distinct enough that it may standalone outside the context of the song and still make (some) sense.
Lyrics (plural) would refer to multiple such units. In general, a single song has multiple pieces: lines, verses, etc. As @AndrewLeach noted in his comment, one would refer to all of the words of a song as the lyrics of a song.
Here are some examples:
Martin Scorsese's documentary No Direction Home takes its name from the Bob Dylan lyric: "How does it feel / To be on your own / With no direction home / Like a complete unknown / Like a rolling stone?"
He loved the lyrics to "Like a Rolling Stone."
He's a fan of Bob Dylan's lyrics.
I would add that in most forms of everyday conversation all that is needed is never referring to poetry unless in educational settings.
"The, that lyric" for specifying one line or stanza of a song
"The, those, lyrics" for an entire song
Lyric can be an adjective or a noun. See ODO for example, reproduced below. Where you found that "lyric is related to poetry" it would be used as an adjective, I expect.
adjective
1 (of poetry) expressing the writer’s emotions, usually briefly and in stanzas or recognized forms: lyric poems of extraordinary beauty denoting a writer of lyric poetry: the lyric poets of Ancient Greece
2 (of a singing voice) using a light register: a lyric soprano with a light, clear timbrenoun (usually lyrics)
1a lyric poem or verse: an edition of Horace’s Lyrics
[mass noun] lyric poetry as a literary genre: stylistic categories fundamental to literary aesthetics—epic, lyric, drama, comedy, tragedy
2 the words of a popular song: she has published both music and lyrics for a number of songs
A publishing house might refer to a particular lyric meaning the words of a single, particular, song, precisely to differentiate it from lyrics in general. But that's a particular usage which might not be evident from a dictionary entry (and isn't really apparent in ODO).
It's definitely an ordinary countable noun, so it's a lyric (meaning a single song) or the lyrics of a song; and you can't *edit lyric because that needs an article before lyric.