Lyric poems themselves are generally classified in three ways: by content, by speech act, and by outer form.
Content: We could classify the poems by Rich and Larkin according to their content, calling Rich’s an autobiography and Larkin’s a love-poem.
Speech act: Or we could classify the two poems according to their speech acts, calling Rich’s a confessional narration and Larkin’s a meditation on estrangement.
Outer form: Or we could classify the poems according to their outer form, describing Rich’s as a poem in unrhymed couplets and Larkin’s as a poem in rhymed tercets.
Each of these classifications requires that we investigate the poem for something different — for what its content is, what sort of speech acts it is engaging in, and what outer form it displays.