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In normal conditions, with the shutter speed, you affect only ambient light, and with aperture you affect both ambient light and flash.
Is this also true when the flash is used in High Speed Sync mode? Or in this case, does the shutter speed affect the flash light, too?
Is this true also when the flash is used in High Speed sync mode? Or does in this case the shutterspeed affect the flash light too?
Shutter speed has some effect with HSS, but for different reasons. With high speed sync, the flash fires many times as the slit between the first and second shutter curtains move across the sensor. The higher the shutter speed, the smaller the slit, and that may require the flash to fire more often in order to create an even exposure. The flash unit has only a fixed amount of energy, so the more times it fires, the less intense each pop must be.
Shutter speed does not affect regular speedlight mode exposure. The speedlight flash is faster (shorter duration) than the shutter speed, so it does not matter (to the flash) how much longer the shutter is open (ambient will be affected).
Shutter speed absolutely does affect HSS flash, which is simply a continuous light same as sunlight, same as incandescent light, etc. HSS is continuous light, for the duration of the shutter open time. Continuous light rules apply.
Other than HSS flash power is rather limited, and range is limited by Inverse Square Law, the camera exposure numbers work exactly the same as for sunlight. If we increase shutter speed from 1/1000 second to 1/2000 second, we lose one stop of HSS light (exposed for only half the time). So we have to open the aperture one stop to compensate. That is of course exactly the same way that sunlight works (simply because both are continuous light, i.e., NOT speedlight flash).
So, specifically, Equivalent Exposures work exactly the same for HSS as it does for sunlight.
1/4000 second f/2.8 is exactly the SAME HSS exposure as 1/125 f/16, like as it is the same exposure for sunlight.
If you have a correct balanced HSS/Sunlight exposure, any other equivalent exposure gives the same exposure result, for both sunlight and HSS. Saying, the one Guide Number applies to any Equivalent Exposure.
Below your Camera's X-sync limit (often around 1/250, what your question refers to as "normal conditions"; any of the Modes (Flash, HSS, HS) is "normal" for some people, so call it X-Sync Limit) a simplified explanation is: The Shutter opens, the Flash fires, the Shutter closes. So the Flash is faster than the Shutter and thus the Shutter doesn't control the Flash's intensity.
Above the X-Sync Limit the Shutter is so fast that the Second Curtain (the closing half of the Shutter) must start to move before the First Curtain (the opening half of the Shutter) has moved; resulting in a moving Slit that travels across the Film or Sensor.
The HSS Flash is pulsed repeatedly to provide continuous (as well as diminishing, and differing color) light for the short duration of the two Shutters (moving Slit) AND a higher Shutter Speed makes the Slit narrower cutting the Flash's power so much that the Flash MUST be fairly close to the subject (when shooting into the Sun the Flash is almost in the Shot.
With HS (the third Mode, which you asked nothing about) the Flash fires a single time, slightly before the Shutter with an enormous pulse (distances of several hundred feet away from the Flash can brightly light the subject) and the electronics times the Shutter to utilize the hottest portion of the Flash. So you're (sort of) back to a normal Flash (not HSS).
HS is what you want to buy/learn IF you're into using Flash.
HSS was developed to use a Flash at higher Shutter Speeds without getting a black bar in the Picture (due to the Curtains blocking part of the Flash) and eats your Battery - HS only eats as much Battery as the brightness to which it is set. Greater investment but more Shots per Battery (for the intensity used).
That was supposed to be an oversimplified (but not incorrect) explanation.
Marc Weiler's Website features many Photos taken with a Elinchrom Hi-Sync, in particular Photo number 13 on this Webpage shows the use of a Flash at a very long distance and a clear view of the amount of light it casts at that distance. Technical Note: You can click on one of those Photos and they open in a Gallery if you don't wish to count.
