I’d like to take a moment to mention that any use of the term “hypnotism” before 1841 is speculative since James Braid was the first to use that term in 1841. Braid adopted the term hypnotism to emphasize the state of the subject, rather than the techniques applied by the operator.
Braid’s technique was a subject-centered approach, unlike the operator-centered Mesmerism, Animal Magnetism or the induction of artificial somnabulism.
In an unpublished work, Essay on the Curative Agency of Neuro-Hypnotism (1842) Baird, referred to “nervous sleep” (meaning sleep of the nerves) by the term “neuro-hypnotism”, and then used “hypnotism” (from the Greek hypnos, “sleep”) for short. Hypnos (’UpnoV in Greek) is the Greek god of sleep. Hypnos was generally depicted in myth and literature as a gentle, benevolent force that brought the restorative gift of sleep to mortals and gods alike.
Braid realized that “hypnotism” was not a kind of sleep; he pursued the use of the term “monoideism” which translates to “single-idea-ism”, but the term “hypnotism”, and its descendant “hypnosis”, have already been widely in use and favored by the public.
The term “hypnosis” (in contrast to hypnotism) gained widespread about twenty years after the death of James Braid. This term gained in popularity first in France where most the early developments in the field occurred.