Re: undertile membranes: There are several types.
One is the type indicated above and is intended to allow the floor and the tile to move horizontally separately. The idea is that cracks in the floor do not cause cracks in the tile.
Another is to provide a waterproof layer under the tile and is used for exterior decks; and it provides the horizontal movement that the first does.
A third is to provide a waterproof layer to retain and direct the water under your shower or tiled tub toward the drain. This used to be done with lead sheet but PVC sheet is used these days. A pan is formed with the PVC and the PVC is glued and/or clamped to a two-stage floor drain. A sloping tile bed is added in top of the PVC using concrete w/o aggregate other than sand. Then the mastic of your choice is used to bond the tile or stone.
Tile cements suitable for ordinary folk have come a long way since the '50s when water emulsion organic mastics were normal practice. These mastics earned a well-deserved disdain: they mildew, they become brittle and the tiles fall off, and they didn't hold worth a hoot anyway. Nowadays cementitious thinset cements are easy to use and excellent. Some are advertised as latex-modified; some you supply the water; some are premixed; lots of choices w/o using the epoxy types. Similar choices for grouts, including texture and color.
I've used some of each of these; I prefer the mix it yourself types. Certainly, I do not prefer the premixed latex-modified grout; what a mess! It leaves a plasticy smear as it dries on rough tile that you really have to work to get off.
I can share pics of installing Granite on shower and bathroom floor, including the PVC and sloping bed. Ditto installing bas relief tiles on a door-sized wall-hung objet d'art.