"Tool room" lathe basically means nothing , other than maybe being abit more accurate in a few things. I think it's a selling slogan such as a "gunsmithing" lathe . I had a 20"x120" Leblonde that was called toolroom . Second operation lathe is a lathe usually with no power feeds such as a Hardinge . Some have cut off capabilities , rear turrets etc . Used for finishing up parts in a production environment .
A tool room lathe is generally a smaller lathe, normally <16 inch swing. As opposed to a larger production machine. Normally has threading capability. Today the lines have been blurred by the marketing hype.
A second operation lathe normally does not have a carriage , normally does not have threading capability, normally does not have a compound. Normally manually operated cross slide with settable hard stops, may have a turret.
A toolroom lathe is going to have better bearings and is going to have better rigidity as well as other nice features. I think a genuine toolroom lathe is determined by the ways being twice as wide as its center height .It also comes with a much better price.
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.
Most of this has already been said here but this is from wikipedia:
A toolroom lathe is a lathe optimized for toolroom work. It is essentially just a top-of-the-line center lathe, with all of the best optional features that may be omitted from less expensive models, such as a collet closer, taper attachment, and others. The bed of a toolroom lathe is generally wider than that of a standard centre lathe. There has also been an implication over the years of selective assembly and extra fitting, with every care taken in the building of a toolroom model to make it the smoothest-running, most-accurate version of the machine that can be built. However, within one brand, the quality difference between a regular model and its corresponding toolroom model depends on the builder and in some cases has been partly marketing psychology. For name-brand machine tool builders who made only high-quality tools, there wasn't necessarily any lack of quality in the base-model product for the "luxury model" to improve upon. In other cases, especially when comparing different brands, the quality differential between (1) an entry-level center lathe built to compete on price, and (2) a toolroom lathe meant to compete only on quality and not on price, can be objectively demonstrated by measuring TIR, vibration, etc. In any case, because of their fully ticked-off option list and (real or implied) higher quality, toolroom lathes are more expensive than entry-level center lathes.
This is an example of an excellent 2nd operation lathe, a small Hardinge 5C collet turrett lathe.
These parts have been turned on a CNC lathe then the V grooves are milled in a CNC mill with an indexer.
The last operation is to part them off from the length of stock used to hold them in the lathes and mills, the blocky apparatus is a parting attachment that is operated by a hand lever.
The long part in one of the chucks is the work stop that controls the length of the part within the accuracy of 5C.
The turret has stops that limit each tool, place each part in the chuck loosely and push it in with the stop, close and part off, you will likely be within .005" if the stock has an accurate diameter within .002".