Thanks to @YaoBoLu and the quoted article: Understanding Nikon's Auto FP High-Speed Flash Sync Mode, I think I get it.

Here's a summary of what I understood.

The key appeared to be in what Auto FP means.

For example, let's say 1/320 (Auto FP) is used. This would mean:

• for each shutter speed lower than or equal to 1/320s, the flash will work "normally", as expected.
• for shutter speed faster than 1/320, the Auto FP mode of the flash is activated.
  And this is the key part here - this means, the flash will fire hundreds or even thousands of pulses of light in a second (will look like a single flash for the human eye). This is necessary to make the correct exposure possible to achieve. But this has its downside - to be able to do this, the flash reduces its power. The faster the shutter speed is -> the more pulses are generated -> the less power per pulse. Which means - the power of the flash is "reduced", meaning - the subject should be closer or will not be lit enough.

What I also understood is, that 1/320s is a bit risky (as it kinda pushes the flash to its limits) and sometimes (often the bottom) parts of the photo may be underexposed.

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This made me think the following - 1/250s (Auto FP) is the optimal choice. Using shutter speeds lower than or equal to 1/250s will allow the flash to work "normally", without the risks of getting underexposed parts of the photo. Using shutter speeds faster than 1/250s (like 1/320, 1/400, etc.) activates the Auto FP mode of the flash.