I think the problem with these disputes over the "meaning" of poetry is that one person talks of denotations and another is talking connotations; perhaps, more specifically, one person is talking extensions (objective qualities for classification) and another intensions (subjective qualities for classification). The reason you, eg, probably don't want to call "lineated prose" "poetry" is because, to you, poetry is more of a qualitative term (ie, speaking to the quality usage of lineated writing) rather than a descriptive term. The problem with all qualitative, connotative, intensionative definitions is that they are much too subjective; they rely on each individual's standards for what constitutes poetry. Whenever you read, eg, famous poets "defining" poetry, they always speak of very subjective qualities (eg, Dickinson's (paraphrased) "I know it's poetry when it blows my head off"). However, if you really get down to denotative, extensional qualities, the only definition that can account for every work we've called poetry since the beginning of the art is Eagleton's.
Personally, I see no reason to be exclusive with the "what is poetry?" question. If someone randomly lineates prose, why can't we just call it "bad free verse poetry" instead of "not poetry" at all? To me, that's why we have qualitative adjectives. I'd much rather keep the "poetry" term descriptive and just apply the relevant negative adjectives when necessary.
As for "verse" VS "poetry," I tend to think "verse" is just metrical poetry. I see no reason to use it for anything else.