It seems to have fallen out of use, but at least a couple of the horse books from the ‘70s I grew up with (A Horse Around the House for sure, and I think A Horse of Your Own but I seem to have misplaced it Invitation to Riding from 1976) described and/or pictured the Tom Thumb pelham.
I wonder if the unfamiliarity among English riders with the Tom Thumb term in that context has to do with age, or where/how they grew up and learned about such things.
Edit: I looked up the previous post I made on this exact topic on another forum and corrected the second book.