The article HS or HSS? What is the Difference? talks about the difference between High speed sync, HyperSync, and Hi-Sync, and explains the difference more clearly than I can here, so go read it. There are some whizzy animations that might help you visualize each one. To summarize:

High Speed Sync: The flash fires many low-power pulses of light to simulate a single longer-duration light, evenly illuminating the subject for the duration of the shutter's traversal of the sensor. Nikon calls this "Auto-FP."

HyperSync: This is something that PocketWizard cooked up. The idea is to fire a single powerful blast of light from the flash, and timing the shutter to take advantage of the brightest part.

Hi-Sync: This seems to be Elinchrom's version of HyperSync.

An important difference between High Speed Sync and the other two is that the whole reason for using HSS is that the flash duration is shorter than the shutter traversal time (the time from the beginning of the exposure to the end), while the other two rely on the flash duration being longer than the shutter traversal. The small AA-battery-powered flashes known as speedlights don't pack a lot of power compared to the larger lights often called studio flashes, and so they have relatively short flash duration. I recall reading somewhere that a Nikon speedlight had a flash duration ranging from about 1/20000s at lowest power to 1/200s at highest power.

Studio flashes, on the other hand, are much more powerful and can have longer flash duration. The point of HyperSync and Hi-Sync, then, seems to be to get the most benefit from the flash by timing the shutter to capture the brightest portion of it.