Defining "engine lathe" is a sure way to get lost in nit picking. Lately, there's been a new thread on the topic somewhere every week.
I have my own prejudices but they are not germane to the point I'm making. If I was to be objectitive I'd say If a particular machine tool passes the duck test for lathe then it's a lathe and let it go at that. The adjective you apply to the noun comes from the uses you put to.
The alleged categories of lathes in similar size classes attract pointless, uninformed arguement. Look at function: If you can use a lathe as an adjunct to your maintenence work it's a maintenence shop lathe,. If you're a toolmaker interested in precision and repeatability, the lathe that fulfils your requirements is a tool room lathe be it South Bend 10K or Monarch EE. If you are working with a bunch of railroad buffs restoring an old steam locomotive you could rightly call the ancient Le Blonde you're using an engine lathe.
A lathe is what it's used for. It's distinctions and usefulness overlap to a degree that arguement over exact nomenclature is not only folly but sophomortic. It's true that some do certain jobs better than others but serious machinists and metalworkers have better uses for their time than pursue rancorous arguement over distinctions without significant differences. Arguement over what constitutes an "engine lathe" reminds me of medieval clerics arguing how many angels could dance on the head of a pin or "I can't define pornography but I know it when I see it".
If it will turn diameters and faces in most metals accurately enough and cut threads, rejoice: it's a lathe.
Nuff said? Go do something purposeful.