From the reading I've done in the past 2 days, it depends on your camera.
Flash sync means the fastest shutter speed which still has the entire sensor exposed at one time. Faster shutter speeds means the 2nd curtain starts closing before the 1st curtain is completely open. Thus if your flash went off, the light from the flash would not reach certain parts of the sensor. (This is a very brief explanation of the subject).
Therefor flash sync is the fastest shutter speed you can use with a flash, any flash. It is a function of the mechanical design of the camera, and doesn't have anything to do with the flash.
From what I understand, a lot of the newer Canon and Nikon bodies, coupled with a newer, higher-end speedlight/flashgun can use a separate type of flash, which will effectively become a strobelight, with many smaller pulses during the exposure. This means that the entire sensor will be exposed to the light from the flash. However this also means you lose the 'action freezing effects' of using your flash, since there are multiple flashes.
Although you understand the processes, your explanation is actually not 100% correct, joshilewis.
High Speed Sync lets you flash at higher shutter speeds than X-sync (the maximum shutter speed that synchronises with your flash when it produces one
flash of light). As typical settings at ISO 800 in reasonably good light can be around 1/2000 sec at f/5.6 the action will be
frozen… not by the flash, but by the shutter speed of your camera. The flash pulses thousands of times per second (@ 50kHz), for a brief period, to produce essentially a continuous beam of light. So, in effect you improve the ambient light artificially for that brief moment while your camera shutter is open, even at shutter speeds up to the fastest your camera is capable of, doing away with the old limit of maximum sync speed. However, both camera and flash must offer this feature.
I reproduced a diagram from Canon that illustrates the point...
Another useful feature of HSS-enabled flashes is that it reverts to normal flashing if the camera shutter speed goes below X-sync for whatever reason. So, setting and leaving your flash HSS-enabled makes sense for wildlife photographers.
Just as an aside... if you get very close to fast action (e.g. insects and hovering birds or hole nesters) you can freeze the action with the light from your flash by dialing down the output to minimum (1/128th on good flashes). At this setting the light flash duration is 1/50000th of a sec! Unfortunately the prerequisite of closeness cannot be overcome because of the feeble light intensity of a flash at 1/128th of its full output.Large view
The image above shows frozen action with water droplets colliding, made with 1/128th output from a Speedlite 580EXII, as explained above.