It depends on the size, design, and lay-out of the "room" or "area".
Such a room might be simply called "a/the study". We would not expect this to be very big, certainly no bigger than, say, 20m x 10m. It would also be identifiable as a room in some way - it would have walls as boundaries and a ceiling that is not much more than 3m above the floor.
A study area could be any size or height, but would be seen as an open area within a greater area or volume.
In my elementary school (grades 1-6) we stayed in one classroom with one teacher all day, and did homework there when the teacher was not teaching something.
In grades 7-12 each class lasted one "period" and was in a different room with a different teacher. A school day was eight periods. Any period that you had an assigned class (and classroom), or you had an assigned "study hall" (and classroom).
These study halls used regular classrooms or the school library. Like any class, you had to be present, and a teacher was there. But there was no instruction. Each student studied quietly, or read. The term "study hall" referred to the assigned class period. It was not the name of the room. We did not have specific rooms for that purpose.