The exercise you saw in the Loi book was Tegatana Awase and it is a precursor to Shotei Awase as far as our basic exercises go.
Just to clarify - precursor as in the order of exercises.
Tegatana Awase - is designed to improve distancing and sensitivity to your partner/opponents movement.
Shotei Awase - is the pushing exercise.
A couple of comments about the latter for those who know. I actually have people do that to a wall when they first learn it since a wall shouldn't move. When doing it another human being proper alignment becomes even more crucial and the exercise can degenerate into something else if both people don't understand what is being worked for. This is a very simple exercise on the face of it but very complex deeper down.