Not quite, think about that. what would happen if the elevators commanded nose down pitch and the thrust levers remained at cruise power! The VS mode commands pitch to hold selected vertical speed and engages the autothrust is speed mode to attempt to maintain the selected airspeed. The LVL Change mode co-ordinates pitch and thrust to make automatic climb or descent at selected airspeeds. In climb, thrust will be limit thrust, (N1) in descent, thrust levers will be at idle. If a speed mode is engaged when LVL change is selected, this becomes the target speed, if a speed mode is not engaged at the time of selection of LVL change, the existing speed becomes target speed.
I have not mentioned reversion modes or protection, in an attempt to answer the original question!