Yes, that is textbook, the pump is failing symptoms. Nothing strange, just textbook pump failure or some sort of indication of a fluid leak allowing the pump to be empty. The pump is moving fluid around when the engine is on and running. When the engine turns off, the P/S fluid no longer moves. If the pump is allowed to be emptied, the next time the engine starts to move aka pump moves, the pump has no fluid and will whine at you because the pump wants more fluid.