The version used by most modern music educators.
Kate's version, which avoids train wrecks and hurt necks.
Here is a PDF of a teacher's guide from the Kentucky Education Network. - with John McCutcheon. You can scroll down to a video where he sings it at around 6:56. Here is a link another video of his section. Here's a link to a wider variety of videos from a similar series throughout the PBS network.
Through discussions at ACEMM's Book Study, I was alerted it was a handclap game.
I found a discussion about it on Mamalisa.com forums that related it to a hand-clap game or rap from 1960s New York based on the Winston Cigarette Jingle. Azizi Powell has also found evidence for this, as shared in this blog post in Pancocojams. In this blog, they mention the oldest known version of the rhyme, but, as Internet Archive is currently under attack, the text can only be found in a more modern text (The Man who Adores the Negro by Patrick B. Mullen). This oldest version was from the 1910-1920s and was written in a short story by Richard Wright.
Here is a thread from Mudcat.org, a digital preserver of folk materials. They also remember it with the Winston cigarette intro.
Below is a PDF of Pennsylvania Folklife Magazine, Vol. 17 No. 1. We need to go to page 16 to find another published version, sourced from a 3rd grade class in Maryland.