re:If a carpet goes from wall to wall, why would anybody talk about wall-to-wall carpeting? Or an area rug? (I.e., if it only covers an area anyway?)

i see carpet as a somewhat permenent covering (wall to wall carpeting is 'nailed' to the floor!)
a large persian carpet, too, is some what fixed.

a rug is smaller.. you might have a winter one of wool, and summer one of sisial, a very small rug is an area rug.

and rug doesn't have to go on a floor.

you could put a rug on a piano, or over the lap of an invalid.

the idea of a fixed floor covering is what makes a flying carpet so magical! --here is an item that is know for its characteristic of being fixed to the floor (eithter by some device, or by sheer size and weight!) and now, its flying!-- its like talking about cold fire!