Technically, a toolroom lathe would have the alignment tolerences cut in half as compared to an "engine lathe", this according to Schlesinger's Universal Tolerences", a book that shows how to test alignments on machine tools and posts amounts of deviation allowed from true alignment. Toolroom lathes commonly have extra features as compared to ordinary engine lathes, such as lead screw reverse on the carriage with automatic stops for threading, relieving attachments ,thread stops for the cross feed, very low spindle speed reduction attachments, and more sophisticated taper attachments, and a much larger range of spindle speeds, and often the use of change gears for diametral pitch threads and special leads.
Second operation lathes are even simpler than a hand screw machine, a small turret lathe with manual cross slide and likely no parallel feed, and a bed turret that may be powered, the second operation machine is made to do work on the cut off end of a part that most of the machining was done on a turret lathe, in fact, both 1st operation and second operation work can be done on a hand screw machine or a turret lathe; the definition of a second operation machine is pretty loose, but generally it does work of a simple nature on parts that have been produced by a more sophisticated machine.