I may be misinterpreting David's post above, and, if so, my apologies. If not, I disagree that water keeps moving after engine shutdown.
Automotive water pumps are almost always centrifugal pumps, which take water in at the center of a vane element and fling the water to the outside of the pump housing by centrifugal force as the vanes spin. Subaru water pumps can put out as much as 52 gpm at 6000 rpm. However, at idle, the output can drop as low as 2-3 gpm. They need to be variable, since the cooling requirements at idle differ markedly from WOT, so the design is a compromise between efficient cooling at high speed and not overcooling at startup and idle. American muscle cars frequently change the pulley size on the water pump to gain rpm if the engine application is hard to cool.
When the vanes stop turning at shutoff, water movement effectively stops. Bernoulli principle, as I understand it, does allow for pressure differentials in closed systems, which may mandate a slight flow, but those differentials quickly equalize with the pump off. Within a few seconds, the system has equalized.
The pressure differential required to pump water uphill can be significant, but only for substantial height differences. In our replicas, we are only talking 1 foot maximum between pump height and the highest point of water circulation, or .43 psi difference, a minor factor. You lose more pressure with hard plumbing bends than height difference.
I may have misunderstood the above post, or I may learn something new if folks chime in, but the above is standard hydraulics as I learned it 50 years ago.