Wrong:
The Map display gets its information direct from the FMC. So the route displayed to the passengers is the actual route taken.
Ur, wrong yourself L337, TopBunk is yet again correct (second time today I've made this comment).
The IFE system on the BA 744 (Rockwel Collins and not very good) does indeed start off with a great circle route, whih is then progressively over written with the actual track.
Edited due poor syntax