The term book press is not an official bookbinding term. It has entered the popular vocabulary to describe what binders would call nipping presses or standing presses.
On a functional basis, any device where the book lies flat on one flat surface while another surface is pressed down on it can be considered a book press. Most "real" presses have a screw on top to produce even, firm pressure. But the crucial elements are the intensity of pressure, however it is produced, and the smoothness of the pressing surface. My first book presses were a little unusual in that they were also my sewing frames.
The book press is used in numerous binding steps. It can be used to flatten signatures when repairing them in rebinding, when laminating boards for covers, and in all steps where the entire flat surface of the book is to be pressed (for instance, after applying cover material).
These are the designs for home-made book presses that I have tried or considered: