This is Texas Poet's quote.
Well yes and no, because certainly narrative poetry can have/show a change in the protagonist and very often does.
Here's what I say in the introduction to my most recent collection:
"Call them prose poems, flash fiction, nano fiction, poetic prose, short-shorts, one-page fictions, creative nonfiction, drabbles (100 words exactly), or call them Carl if it suits you. To me they all mix and meld, intersect, overlap and combine into something I like to think of as Prosthetic Amalgams.
Kenny A. Chaffin – February 2015"
I think truly it is somewhat dependent on what the author says. In reading 'Great American Prose Poems' edited by David Lehman he discusses this a bit in the introduction but also includes selections that truly could be either one.
To me Prose Poetry tends to be a bit more explicitly poetic in word choice and in phrasing as well as using poetic techniques alliteration, onomatopoeia, rhyme and etc. but not always.
I began studying flash non-fiction a couple of years back which led me (back...I'd actually written some flash decades ago) to flash fiction and prose poetry and in that process I kept running across pieces (such as The Deck
by Yusef Komunyakaa, Leap
by Brian Doyle) which were published in different places/collections as flash or poetry. Also works by Sherri Flick, Lydia Davis, Stuart Dybek and others.
In truth there is a spectrum and no clear dividing line, best just to read and enjoy!