I have no idea if the terminology has morphed over time, but precision used to refer to tolerances of .001, high precision meant .0001, and super precision was .00005 or less. At least that's how the machine and tooling grades were explained to me.
With lathes the differences are mostly in the spindle runout and the straightness and parallelism of ways, and whether and how the ways are hardened.
Bench lathes are self explanatory, tool room are generally more rigidly built and to tighter tolerances, machine shop can be as robust as toolroom but not built to as close tolerances. Gap bed are just that, they have a removeable bed section just in front of the headstock to make room for turning large short pieces without having to have a much bigger lathe.
As you noticed, a lot of the differences aren't visible to the eye. You can't tell by looking what class bearings are in the spindle or whether the bed casting is extra beefy or has high precision ways.
Second operation and turret lathes tend to have small fast high precision spindles in relation to their swing, and most use collets for the majority of their work. They also usually have infinitely variable spindle speed and very short working travels compared to the bed length.
There's a whole lot more to be said on the topic, but that's enough maunderings from me. The current pros can flesh things out and no doubt correct me on some points
Tiger