I was told by an old old timer the engine lathe was a lathe that could produce an engine so had to be screw thread cutting, capable of key and flat machining, boring, and turning tapers.
I know some lathes are called Engine Lathe with not being able to turn tapers..but they do have taper attachment available.
So the notion that the term came from...
QT gvasale post #5: "I heard once that it was the type of lathe James Watt used to build his steam engine" seems logical.
Someone might search to find if the first use of the term "Engine Lathe" came from this time.
James Watt (1736 to 1819)
Just my opinion here like every other opinion.
Next, we might question what is an engine? Could a windmill or a water wheel be called an engine?
what is an engine - Search
Wind engine Definition & Meaning - Merriam-Webster
Is the word "engine" limited to fuel-burning engines only?
*I am just throwing this out for thought / I don't wish to argue about it.