This is a tough question.
Eliot, in particular, demonstrated the primacy of meter over rhyme in Four Quartets. (Rhyme is generally considered to be the primary mnemonic device, a critical element in that poems are traditionally meant to be recited, but that particular poem shows that meter alone serves the function quite well.)
Rap, the most popular and vital form of modern poetry, heavily utilizes rhyme, but sophisticated meter still seems to be the distinguisher of the best work.
After Duchamp and the Beats, a lot of people seemed to decide poetry was anything one called poetry, which led to a lot of very poor "verse", but I'd make the case that even free verse, if it's any good, has impeccable meter.
Maxwell Anderson was quite unequivocal on the subject of form:
"It is the fashion, I know, to say that poetry is a matter of content and emotion, not of form, but this is said in an age of prose by prose writers who have not studied the effect of form on content or who wish to believe there is no limit to the scope of the form they have mastered."
Source: Maxwell Anderson, "Poetry in the Theater", 1939 | Excerpt from American Playwrights On Drama
My personal feeling is that poetry is defined by form more than content, because even doggerel is a low form of poetry.
I'd say that Wikipedia extends the definition a bit too far with the requirement that poetry "evoke meanings in addition to, or in place of, the prosaic ostensible meaning" because it's quite possible to write perfect verse with no deeper meaning.
You might find Robert Graves' definition from the White Goddess to be of some interest:
"...language of poetic myth anciently current in the Mediterranean and Northern Europe was a magical language bound up with popular religious ceremonies in honour of the Moon-goddess, or Muse, some of them dating from the Old Stone Age, and that this remains the language of true poetry..."
Source: The White Goddess
A little bit out there in a modern context, admittedly, but it's important to remember that the Festival of Dionysus was a religious event centered around dramatic poetry, the Homeric Hymns has religious function, and the pronouncements of ancient oracles were highly stylized language with obscured meaning. ("Stylized language with obscured meaning" is one definition of poetry.)
As a sometimes poet, I find the idea that poetry is a "sacred language" to be quite resonant, and substantiated by the sublime effect great work produces. This is to say the effect of great poetry goes beyond the strictly rational, which is the domain of both creative insight and religion.
It is not an uncommon idea. A second excerpt from Anderson's essay heavily utilizes religious language:
"I believe with Goethe that dramatic poetry is man's greatest achievement on this earth so far, and I believe with the early Bernard Shaw that the theater is essentially a cathedral of the spirit, devoted to the exultation of humanity, and boasting an apostolic succession of inspired high priests which extends further into the past than the Christian line founded by Saint Peter."
Source: Maxwell Anderson, "Poetry in the Theater", 1939 | Excerpt from American Playwrights On Drama
Anderson extends the metaphor in the subsequent passages, notably using the term "mystery" in the religious context:
"It has been, even at its best, a democratic temple, decorated with more gargoyles than saints, generously open to wits, clowns, excoriating satirists, false prophets, and crowds of moneychangers with a heavy investment in the mysteries. Lately it has recognized the mysteries only as a side show, and have been overrun with guides who prove to an eager public that all saints are plaster and all prophets fakes."
ibid.
Anderson was commenting on, in part, Shaw's later attack on Romanticism in drama, and Shaw did win that argument as far as the 20th Century is concerned. (Anderson's most famous play, The Bad Seed, was a prose drama;) But I like to think Anderson might have felt vindicated had he lived to see the monster hit Hamilton. Poetry maintained a strong presence on the stage in the form of light verse in American Musical Theater in particular, which is a significant form.
Shaw's direction brought us to the pinnacle of theater in the 20th century with playwrights like Beckett, but it's unclear that greatest modern dramas are entirely "prosaic":
"They give birth astride of a grave,
the light gleams an instant,
then it's night once more."
SOURCE: Waiting for Godot
Here's an excerpt from the "Was I sleeping?" monologue which is hard to regard as not being poetry. David Mamet is well known for incorporating an underlying metric structure in his prose plays, which can rise to lyric heights. We tend to think of examples such as these as "elevated language", referring to poetic qualities of language embedded in prose contexts.
Graves makes a point that poetry evolves over time across different cultures, and in different periods "high" and "low" forms are in or out of favor.
This highlights the difficulty in defining an art form which, at it's best, uses language to convey the ineffable.
Finally, it's worth mentioning that the poet Nicanor Parra, in seeking to break free of the conventions of the past, coined the term "anti-poetry" to distinguish the new form and approach. But even anti-poetry is a type of poetry.
The forms of poetry are vast and ever changing, but all poetry shares a common trait that may not be possible to put into words.
