In the first FH video (from a month ago) you transfer the energy from the ground, leg, hip, torso, arm, and the stroke has spin venom. All is there. But it feels like the arm is moving in synchrony with the torso. As the torso is rotating, so the arm moves with it. It appears you add some forearm muscle driven acceleration, which occasionally results in a bit elevated shoulder at the end.
In the second FH video the start of the chain is the same, you let the torso (and things before) rotate, but start to move the arm a tiny bit later. The torso acquires some momentum in advance (primary acceleration) and as TTNuri put it, this momentum is transfered to the arm via shoulder joint. Now the arm already has some momentum, and you don't need to add so much more. You can let it go. As a result, you can use less muscle and be more relaxed. You have let the body do the (some) work for the arm.
In the third FH video, you do this in Falkenberg, which is different level. Although it looks simple, it isn't, imo. I try to do it too, but on robot only, we here can't play with people, and it is uncertain when we can again.
With BH, I can not say much. What I noticed, is that my shoulder is more strained from BH spin than from FH spin now. So I am now focusing more on being lazy, let the body do the work. I have the best feeling when I also feel the hip rotation there too, and pushing the left foot. I am not focusing on it, but that is when I have the best feeling. You also do this counter-move with free-arm, which Der_Echte mentioned recently too.
I think children learn more linearly, they start, and linearly are instructed to add more and more speed, and it goes with their growth. For the adult learners I think more non-linear learning is better. Do the movement in various arm-speeds, slow, middle, fastest. This works as error-correction, you can't really move the arm, both relaxed and truly fast, in a wrong way. So if you try, both relaxed and fast, the body does. It may not work the other way, learning only slow, and hoping you can speed up later. You may have learned something what you simply can't speed up.
I am looking forward your explanation and total refutation of my BS