It is the vacuum check valve for the vacuum hose from the manifold into the booster. When operating correctly, it should only allow air flow from the booster to the manifold in order to create a vacuum on the piston/hydraulic side of the booster. In this case, the interior valve has disintegrated and allows air flow in either direction.
In this vehicle, the check valve is extremely easy to remove and test. Locating the hose from the manifold to the booster, the check valve is the plastic 90 degree connector into the booster; there are no clamps so simply pull the check valve from the rubber grommet in the booster and then pull the valve from the hose.
Air should only flow from the booster side to the manifold, so blow into the manifold side to test. If air flows in this direction, it is not working. This was a nine dollar part from NAPA for me.
So why does the pedal lift? The manifold is maintaining vacuum while the engine is running, but as soon as the engine stops, air flows back into the vacuum side of the booster and I loose all the assistance I was using to keep the pedal at that level. Unless I apply a great deal more force, the pressure on the brake hydraulics will overcome the pressure from my foot, raising the pedal.
This check valve provides a degree of safety for the case in which the engine stalls while driving. When the valve is working, vacuum is maintained even after the engine is shut off, allowing for at least one vacuum assisted stop of the vehicle. After the vacuum is gone, the driver will need to apply far more force to engage the brake effectively.
Most explanations of vacuum brake booster do a lot of hand waving. The only complete description that I could find of the valving and how equilibrium is achieved was in this youtube video. Once I saw this, it seemed there could only be this one (single) reason.
Thanks to @GdD for helping me think this through.