This is almost right. If the turn is to be level, elevator must begin to be applied as soon as the wings begin to bank and increase until the maximum bank angle is reached. Up elevator is held as long as the airplane is turning. When rolling the wings level, up elevator is reduced as the bank is reduced so that when the wings reach level, up elevator is no longer required.

Up elevator doesn’t turn the airplane, it is only necessary to maintain altitude for a level (or climbing) turn. The airplane will turn anytime the wings are banked, however if up elevator is not applied the airplane will also descend.