Someone asked me the other day about writing poetry or lyrics for music (because they're mostly the same thing, right?) and hoo boy I have rarely felt so in over my head. This is definitely not my forte, so be sure to take literally everything I say on this subject with a main course of salt. A few grains won't do it on this one, my lads. It's time to go off the deep end.
So to start, there are rules about what qualifies as certain kinds of poems, like sonnets and haikus and things, but there's nothing in God's great green earth that can tell you that something isn't a poem. You can write anything - it can rhyme or not rhyme, it can be long or short, it can have a rhythm or none at all - and if you say it's a poem, there's not much anyone can say to argue that point.
The only thing that differentiates poetry from its companion, prose, is that poetry tends to be more symbolic and flowery than prose, but I've read some extremely prosey poems, so... what'cha gonna do?
And maybe that's too broad a definition, but there's not a lot you can do to narrow it down without trying to fit poetry into a specific form or genre. And to be clear, poetry can be in any form or genre - there are even whole novels written in poetry - but the minute you try to tell someone that something they wrote isn't poetry, you're stepping into very dangerous territory.
It's safer (in general) to accept the author's definition of a work, especially while they're in the same room with you.
Now onto the subject of lyrics. I need to point out that while a poem and lyrics might look the same on paper, they're not the same in reality, because lyrics need to be sung, and that requires a process called "text-setting," which is the process of placing words into the context of the music so the lyrics can be sung smoothly. Some musicians do this for their own music, and that's awesome, but lots of professional musicians need to hire someone else to do the text-setting for them.
Someone who writes the lyrics to a song may very well be a poet as well, and that might make things easier for them, but we call someone who write lyrics a "lyricist," because they may never wrote poetry, and that's okay.
Again, it's the definition of the work by the author that matters, not what the audience thinks it is - at least as long as the author is present, anyway. What you say when they walk out of the room is up to your own judgement.