A half bound book is not one which is liable to fall apart because nobody completed the job: it’s a book which has a binding with leather on the spine extending onto the front and back covers, and leather corners. Leave out the leather corners, and you have a quarter bound book. There’s even a style with very wide corners which rejoices in the name three-quarter bound.

Antiquarian book dealers often get even more specific, saying half morocco or half calf for instance.