I think your assessment is correct, I am not trained in either physics nor piano tuning, but I do have some understanding of how these mechanisms work as an engineer. I think your assessment that it is the interplay of the two vibration modes interacting which gives that phase shift we see in the oscilloscope. I would rather say you'd need multiple microphones at different places in different orientations to capture more points of sound source, but that still cannot capture the state of the air as it is vibrating, I'd expect continuous shifting rather than the undulation you're seeing, the modes will come and go, interfere with each other constructively and destructively giving the result we see.
Yes - this seems to be plausible. It's more complicated as Bechsteins often display "false" self beating strings which indicates two frequencies at once and I wonder if they did that deliberately as it's so common with all their instruments. What I took away from this experiment is that mainly tuning to the fundamental frequency rather than combinations of harmonics one's got a significantly reliable signal wherever the mic or machine is placed. Perhaps some modern apps or devices can be too clever in looking at a spread of harmonics as the effects of thick stiff wires giving inharmonicity will play games among the harmonics.
I think the wedge shifted around the 3 minute mark. The note is jumping to the third octave, and decaying too quickly. Should you have muted the highest keys? Just my two cents, nice video.
I would never have done the experiment were it not for some theoretician to swear blind that there was a variation. On the fundamental note I think one's hard pressed to see a variation. The soundboard should convey the vibrations from the bridge and distribute them. I normally place this ETD device within easy reach, either next to the strings I'm tuning near the agreffs or further back over the dampers or on the strings behind and move it only if the signal shown by the dots is too weak. I've never worried where I've put it as far as getting the right pitch is concerned.
