However, I will also mention that contrary to the video, most DSOs have absolutely no difference in behavior between your #3 and #4 cases, since the trigger signal is branched off after the input coupling; if the signal is already AC coupled at the input, AC coupling it again at the trigger does virtually nothing. Hence why most DSOs default to "DC trigger coupling", which could be more accurately named as "triggered off the signal you see on the screen".
There is more to it because if they were triggering off of the literal signal seen on the screen with DC trigger coupling, then changing the vertical position might be expected to change the trigger position; oscilloscopes where this happens support AC triggering or something more sophisticated to avoid this behavior which otherwise looks rather peculiar.
Modern DSOs use digital triggering so the vertical position adjustment can easily take place after the point where the trigger is processed however DSOs which support offset adjustments have this problem; the DC offset will change the trigger point unless AC coupled triggering is used.
Some oscilloscopes combine the vertical position and offset adjustments into one control since the later can do things that the former cannot. Rigol apparently lacks an offset control despite their misleading documentation implying that they do have one since adjusting the vertical position has no effect on the trigger point when DC coupled triggering is used.
This is hard to word well, so bear with me. Isn't it possible that the software just looks at the vertical position control, commands the offset circuitry accordingly (this must happen surely, or else you'd need more than 8 bits of ADC to get 8 bits of output), and then it calculates which digital value corresponds to the chosen trigger level and sends that to the digital trigger circuitry? That way, a change in vertical position will change the both the digital trigger level, and the values coming in off the ADC in exactly the same way, so that it all cancels out and you end up triggering at the same points in time.
FWIW, I was just playing with my DS2202 and moving the vertical position causes the trigger level indicator to follow on-screen, staying on the same voltage-at-the-input, as you would hope. However, if you thusly move it far enough off-screen, it clamps and you can see the trigger level voltage (also shown numerically on screen) changing as it reaches values outside the range of the ADC. Specifically, at 20mV/div, and with the vertical control centered (such that the 8 vertical divs on screen cover the +/- 80mV range), the trigger may be set anywhere from -100 mV to +100mV, as if the ADC range actually covers 10 divs. With the vertical on-screen range extending from -120mV to 40mV, the available trigger settings are -140mV to 60mV. Always an extra div available at the top and bottom of the range. So that seems to confirm that the trigger level selection is converted by the software into a corresponding digital (or at least, post attenuation+offset in the analog domain) value, and used accordingly. This seems like ideal behaviour to me, modulo the limitations in the range of available trigger level voltages inherent in doing triggering post-ADC.
BTW, you're right that my wording was perhaps deceiving -- all I meant by "what you see on the screen" is just referring to the idea that the trigger point can be rendered meaningfully as a horizontal line on-screen w.r.t. to the trace.