I'm no expert but I think the door lining is the frame of the door that the hinge attaches to and is shaped to stop the door flush to the frame. A door casing is the decorative architrave that surrounds the door - at least that's what wikipedia said anyway. Hope this helps.
in this country the terms door lining and door casing are really quite interchangeable. go to jewson's you will buy a door lining. the same thing in wickes and it's a door casing.
and strictly speaking the term the 'door frame' actually refers to the structural timber opening in studwork/brickwork that accepts the door lining/casing.
there is similar confusion as to what constitutes a door frame, door lining and door jamb! these days all seem to be quite interchangeable. even professional chippies and joiners don't seem to be able to agree on which is which!
in the US the door casing is the architrave which is perhaps the reason for the abiguity.
A door lining as it suggests is a lining of 3/4" timber with a plant on stop forming the rebate. A casing is the same thing but the rebate is machined out of a solid piece of 1 1/4" timber, forming the 1/2 inch rebate.Yes the side of the frame is called a jamb,but usually a door jamb is the side rail if you like of the door