The theory that the word derives from the American author Edgar Allan Poem (the “m” is silent) has been completely debunked as “fake etymology”. A substantial majority now accept that Poe was born in 1809, when the word was already well established in the English language.
Nevertheless, social media sites continue to be filled with the rage of those who cite YahooWurdz.com as their source. Things are heated, and we don’t expect the controversy will die down anytime soon.
I had never heard that folk etymology. It’s funny, but definitely not true. Ultimately it is from the Greek word “poiēma” meaning “work, creation” and later “written work” and specifically, one in verse. It entered Latin from Greek, then French from Latin, then English from French.