Some of it is, some of it is not with respect to where a hit ball lands and whether or not it is considered fair or foul.
The exception is a batter struck with a hit ball while still in the batters box results in a foul ball regardless of whether they are in the fair or foul part of the box.
If a ball settles on the back part of the plate. Is that fair territory?
The plate is entirely in fair territory. Connect an imaginary line from each foul line to the back corner of the plate. The portion of each batters box inside the foul lines (closest to pitcher) is fair and the back portions are foul.