You seem to imply that the book from which you study does not DEFINE what a Möbius transformation is. Then discard this book and choose another one. Unfortunately the terminology is not settled. The standard definition of Möbius group (in any dimension) is the group generated by inversions in circles.
According to this definition, in dimension 2 , $1/\overline{z}$ is a Möbius transformation. The subgroup consisting of orientation-preserving transformations is accordingly called the group of orientation-preserving Möbius transformations. In dimension 2, it consists of linear-fractional transformations.

However many authors restrict the name Möbius transformation to orientation-preserving (conformal) maps, and this creates a confusion.

On my opinion, orientation-preserving Möbius transformations of the plane should be called linear-fractional transformations.