In the songbook Folk Songs of Canada, 1954, edited by Edith Fowke, the version of "Donkey Riding" is that of the Oxford Song Book.

Fowke's notes on the song echo Wood/Oxford, plus some of her own suppositions. She states, parenthetically, "(The 'donkey' refers to donkey engines used in loading cargo.)"

My present guess is that this is the source (handy for Canadian kids'/school singing, the channel through which the song was most popularized) of the popular idea about the steam engine metaphor.