Thanks Navin, for the "Follow-Up". In the first video about E-guitars, he got rid of the neck and the body. Didn't matter that much. And I understand it: It is the string vibrating and the pickup receives the vibrations. I think, the wood has some impact on the vibrating string, but is so little; basically negligible. Anything else has more impact; even the temperature of the air around the strings.
But speaking about acoustic guitars, the wood - especially the wood of the body - acts as the amplifier. There aren't magnets and coils, picking up the vibrations of the strings before the body has any impact in sound. About the pickup test in the very end: Those acoustic guitar pickups work entirely different from single coils and humbuckers. Acoustic pickups don't pick up the fluctuations in the magnetic field, caused by metal strings. No, they pick up the vibrations of the body itself. No wonder, you could hear a huge difference in tone.
What does this mean for pianos?
Yes, the wood of the soundboard matters. And the wood of the rims maybe too, but to a lesser extent.