With the lower legs being planted in place (not moving) this is somewhat akin to leg extensions, but not quite. Not so much a actual squatting motion. But the name Roman Chair squats can still apply.
Getting a great more focus on the quads, excluding the hip and glutes (almost) with this movement. Also getting a much better stretch, as with the Sissy squat (a much over looked very good quad exercise). Roman Chair squats can be an exceptional exercise to bring out the full potential of the quads, but learning to do it the correct way will be the prime factor. The picture give a good example of the correct way. Some have added DB's in both hands, holding a 45lb (usually) plate/DB against the chest or a BB as if doing front squats. The old timers held a bar over head when doing these (real strongman stuff). Don't really need a lot of weight, or no weight at all, to get good quad results. RC's also work the abs. Could even do sit up's from the bottom position on a RC.
The equipment, shown in the pictures, is a nice example of a true Roman Chair. Some of the things that are past off, and want you to buy as Roman Chairs, are just poor attempts at the real thing. In high school, a friend made a version of a RC, a fine and useful piece of equipment. First thing I noticed was the stretch on the quads. Some of the guy's were really bring up the quads very nicely in a somewhat short period of time. Can make one with wood. Or if handy with welding/cutting, one from mental (pipe and sometimes angle iron). Some adapt to things in the gym to meet their needs.
Good Luck
Good Luck.