Are these terms also accurate for a camera that has a leaf shutter or no shutter?
A leaf shutter will sync at all speeds, so most of the terms don't really apply.
Dragging the shutter really relates to allowing more ambient light contribution to the total exposure. As such, it could be anything slower than the maximum sync speed.
Are these terms accurate for the times when the camera performs the sync function as well as for the times when the photographer manually performs the sync function?
Nope.
Are there other related sync terms?
Although most modern cameras and lenses now only perform X sync (0 delay) for electronic flash, don't forget M sync (slight delay for flash bulbs) available on some older lenses.
If you turn off the flash, and then fire the residual charge, would that be called "draining the sync"?
Other than balancing the ambient light in a scene with the light output of a flash, are there other uses for synchronizing a flash to fire with slow shutter speeds?
Sure. In part, the balance between flash and ambient light depends on which you want to be the primary light source. You might prefer, for example, for the flash to provide only fill. Using a slower shutter speed might also be used to introduce some motion blur into the image. One might also use a slow shutter speed in conjunction with flash bulbs, if the camera or shutter doesn't have M sync.