Cold heart (song lyrics)

[busker]

Song lyrics are written with a certain tune in mind,
Bernie Taupin, amongst others, might disagree.

and nobody would know the first thing about Bernie Taupin if it weren't for Elton John. 
Elton John, OTOH, could sing the lyrics of 'God Save the King' and make a big deal out of it.
A lyric without a song is just any ordinary poem.

See my point about Schubert and Heine.
You can set ANY poem or piece of prose to music.
From wordy Gilbert and Sullivan and Noel Coward to grunty Aerosmith.

EDIT - actually, with singer-songwriters, the words are made to fit the tune. With singer + songwriter partnerships, the words and the tune are developed independently, or with the words coming first. During the creative process, the composer thinks of a tune sans words. That's how I understand these things work for the most part.

Lyrics of the first kind / Lyrics of the second kind
I take your point, but how can one distinguish between the two (when a lyric is posted on the board?)

You can't. You can only crit a lyric sans the music as a poem, and it would be subject to the same set of rules as a poem.
Under those rules, Shelley's Mutability will be a fine piece of craftsmanship, but Mcartney's rhyming 'Oh no, Oh noooo' with 'I say go go go' will be seen as desperation.

With the music, everything changes. 
Pedestrian lyrics will do well enough with the right music. Witness 'Like a rolling stone'.

There's one more thing: the English language is not the best suited for songs. It doesn't have the 'roll' of other, more poetic languages.
It is very well suited for contemplative prose and poetry without music.