It depends which book!
The ISTD syllabus, which was based on work done by Alex Moore generally uses the overlapping step method which describes the 4 step feather.
Guy Howard, in his book which is the IDTA bible, removed the overlap and just uses the 3 step feather. That is the book I prefer, possibly because it was what I learned first, and I generally teach that way. When I did my exam using the ISTD book I kept forgetting the "extra" step.
Bear in mind the the books are not intended as dance instruction books. They are the bible against which a teacher is examined. I think this is why the extra step is used because it enables the student to demonstrate those aspects of a figure that effectively form part of the next step, such as lowering and completion of footwork, completion of body turn, correction of sway etc etc.
One interesting effect of the different styles of the two main text books is the description of the foxtrot three step. One says it is LRL and the other says RLR. If you ever get two teachers from different camps together you could try innocentlay asking "what's the footwork on the second step of a three step" and wait for the arguments.