Hyper sync - long duration flash over the entire curtain movement
True high sync speeds - standard sync method but @ above 1/250th. (Which is still the best and most efficient but expensive.)
HSS - pulsed flash over the duration of the curtain movement.
I think you misread my comment, all of these methods are ways of syncing at High speeds, hence HSS but the most true form is the traditional method and the others are a workaround.
There is nothing 'true' or 'high sync speed' when sync'ing above 1/250th or 1/180th or even 1/60th. HSS is sync'ing multiple bursts over a curtain movement - where the sync is with the slit being exposed is perfectly timed with the speedlites. Sync'ing with a Leaf shutter doesn't involve multiple bursts.
Sync'ing with a 'slow' strobe or speedlight where your shutter speed is faster than the lights t.1 time is just that, dealing with slow lights. The issue is that you're not going to get even exposure over the frame, let alone between shots.
Now, with all that said, with the faster studio lights, like Einsteins in the fast mode, or the Bron Move2 packs that'll do a 1/10,000 flash duration, you can 'sync' at 1/125th or even slower, and 'freeze' your subject with light. Then, all sync'ing at a faster speed does is allow you to kill the ambient light.