Basically the two types of pump that are appropriate for this application are the centrifugal pump and the displacement pump. Both types can be submerged.
The displacement pump uses small chambers built into a rotating disk. When operating, the action creates suction on one side, often allowing several feet of water to be LIFTED and ejected. Displacement pumps such as the FloJet series do not need to be submerged and can be located some distance from the liquid.
The centrifugal pump requires fluid to fill the vane chamber, without air. These work by spinning an impeller at high speed, creating a pressure difference between the fluid at the centre of the vane and the outer edge. The low pressure at the inlet draws fluid into the chamber, but any air present will disturb the vacuum. Centrifugal pumps generally have a higher flow rate and operate at much lower pressures than displacement pumps.
So for the water dish, if you can guarantee that there will always be water for the pump to work with, the centrifugal pump would be least expensive, quieter and reasonably reliable. If the dish runs dry, the pump could become damaged, so a method of detecting the water level would be required.