acheo your observations are correct but use of the words "opposite direction" to describe the initial movement if the FPV is wrong. As someone said earlier, when u initiate a change in pitch or go off wings level into a roll, the aircraft takes a fraction of a second to actually change direction.
So take the "steep turn" exercise in the simulator. upon initiating a turn to the left, the fpv continues going straight momentarily with inertia but will start changing heading very soon. this delay is visible on the pfd by a slight increase in the gap between the aircraft symbol and the fpv. this isn't because the fpv is going away from you, but you are going away from the fpv.
during the turn, the fpv doesn't come to the centre of the pfd because there is a slight lag which is caused by the minor but constant difference in the direction of the aircraft longitudinal axis and the direction that the aircraft is actually flying.
however, when you stop banking and return to wings level, you will find that the fpv returns to the centre of the pfd. Or as per your observation, goes towards the direction of the turn. (no crosswind conditions of course).
So to sum up, you are going away from the fpv and it is trying to follow when you initiate any manouver.