Classically speaking, these two definitions are equivalent. However, they are not in a discrete geometry. If, e.g., the distance function is defined as the lowest number of hodons that connect two given points, then the two definitions are not equivalent. Using definition (a), the circle will have the shape of a square (a well-known fact in so-called taxicab geometry) and thus useless to define adjacency as done above.  Definition (b) on the other hand produces a figure that can approximate a Euclidean circle as close as one likes. In that way Forrest’s definition for adjacency is acceptable within a discrete framework, as no reference is made to a Euclidean distance function.