Refering to my earlier post, "Shop Theory" 5th edition, by James Anderson and Earle E. Tatro, copyright 1934, has the following: " The father of the modern engine lathe was Henry Maudslay who first invented the slide rest." He also combined the slide rest with the lead screw with change gears. " This took place in the early 19th century and made the the lathe the most important machine in the industrial revolution, for without the lathe James Watt's steam engine would never have been built. Because it machined the parts of Watt's engine, it became known as the engine lathe." Other sources simply state the engine lathe to be powered by a engine, whether it be gas, electric or steam. Now, we all know steam was the first type of these three power systems.
Also, how the term "engine" a term to denote a mechanism, I haven't had time to check, but Eli Whitney's invention of the cotton gin in 1796, may have a clue. FWIW, Webster's says the word "gin" a modification of the Old French "engin" dates to the 13th century, and means "any of various tools or mechanical devices.Incidentaly, Author James Anderson was apprenticed into the trades in London, England.