It depends on many factors but IMO bathrooms and mud rooms do not need wainscoting at all. You also need to post a bit about your style since this type of trim is IMO pretty formal and often has an adverse effect on hanging art. This can be seen in the DR with the pale green walls in your idea books . The wainscoting is so high the art can only be hung too high.

The pattern that I've seen in older homes from the late 1800's and early 1900's is the wainscot in the dining room is often different from the rest of the house. The dining room wainscot is usually higher and often includes a plate rail at the top. So, if it can be different there why not in bath and mudroom.

We struggle with using different paneling (raised and recessed) styles in the same house. Appreciate other’s view on more formal rooms having raised paneling/wainscot while less formal rooms have simple recessed panels/wainscot.