Part 2.
After you get a sense of vibrating strings and sound, we can look at the damper part of it.
If you pluck your string to vibrate freely, producing a sound, and you drop your other hand (finger) on it, the sound will suddenly stop. Put another way, you have stopped it from vibrating. In the same way, if you are holding the dampers off the string via the pedal, and you release the pedal, the dampers fall on the strings. That stops vibration and sound. This is the crudest way we learn to use pedal.
Pluck your rubber band again. (A thicker band stretched rather tightly works best, since it will vibrate more strongly.) This time, while it's vibrating, try to bring your finger near it just so you can feel the vibrating sort of tingling against your finger, or like the sensation of a mosquito's wings or a puff of air brushing past you. It's a delicate "almost touch" getting close to the string. Since it's vibrating side to side, you're just catching the outer edges of that. You might manage to semi-stop some of the vibration, so that it still vibrates, but less. I ended up with a kind of weird buzzy sound. ........ So if you have played a note with the pedal down, fully lifting the dampers, and you raise your foot just a fraction inside a sweet spot, you might stop some of the vibrations / sound that is already going on, and get it to be a bit quieter.
My teacher talked about this, and also the video you referred to. That teacher is playing an expensive, top Steinway grand piano, meticulously maintained, with everything balanced perfectly. Most pianos would not have that degree if sensitivity and fine balance in its mechanism. (This, from our conversation).
2. Still along these avenues. I suppose that if I had a hair's breadth kind of distance from the rubber band before plucking it, then it would start to vibrate, but not be able to reach the full side-to-side due to the interference of my hand, so that would change the initial sound. That would be like the pedal being half down, doing an almost-lift of the dampers almost off the strings. This is a stupid way to try to play quieter, since you can do so simply via the fingers. It might have an effect on quality of sound, like maybe more buzzing or so. That's probably the "half pedal" idea. This is how it would work.
The limitation of the piano is that once a sound is started, it cannot be altered. There are pedal tricks to try to alter volume of a started sound. So if your damper can "partly stop" a sound as in example 1, maybe you can make it go suddenly more quiet.
My teacher gave me an idea, with the key word "quick", but I think is "quick and light", where you give the string a quick and light touch. Imagine your rubber band, which is vibrating, and you semi-touch it for a milli-second, slowing the vibration but not stopping it.
I have a screenshot of when I first tried this and too extreme results. You can see the sudden drop in sound (the fat squiggles, become skinny squiggles, like a tadpole with a tail).
https://www.dropbox.com/s/730wk932pq751xu/fast%20pedal.jpg?dl=0This is the sound that went with that screen shot. Later when I play two notes in a row, the idea was that the first note continues, but "less", so that it is still heard. like in a waltz, you want that first base note to be heard: your LH can't stay down there since it has other notes to play, but you also don't want the whole thing to be a blur.
https://www.dropbox.com/s/br6e9kq1wv2p5fm/20.01.24a%20fast%20pedal.mp3?dl=0A few days later after working on my teacher's feedback I refined it. The tadpole has a fatter tail. Here where I play 2 notes, you want the first note to still blend into the second note, but it's been partly damped. This "quick and light" pedal was employed.
https://www.dropbox.com/s/z47mdo7u4uvosps/20.01.27a%20fast%20pedal.mp3?dl=0