I am from a building background....so....a batten is something fixed on its own, like a piece of wood say screwed to the wall before hanging say cupboard onto the top edge of it. Or you can have a batten fixed onto a floor..so a single piece fixed in its own right. A cross peice I would say is any piece which crosses across something and could have a more specific name also used. A rail can be a runner beneath a table top or worktop, could be a face piece along the front edge of something ie below a window, or rail as in a fence running horizontaly. But could be more specific as in top, mid or lock, and bottom rail of a door, or say window shutter board. Stretcher I would use as in something which stretches across something as in say as 'batten' between two gables or haffits. Could also be a stretcher below a worktop. Stile is as in the 'rails' which are vertical and either side of a door make-up as in hinge stile or lock stile or closing stile, or.....confused? Stile is probaly the easiest one to define. In the joinery and building world a lot of names were used as in the body parts. Mouth, ear, birds mouth, cheek, sole plate, heel, toe, nose, foot, elbow, and there must be more around or were all used. More thought required here...nearly confuse myself now.
Berst wishes..this could be a long subject with interest. Spelling now corrected.