Yes, soundboard is the right word for that part of a piano.
By the way, I own and use a pair of these bone conduction headphones. They're not exactly meant for music with much base, and they're not going to provide extremely high audio fidelity, but I enjoy them for listening to things while outdoors. I can listen to podcasts, music, or just the GPS's spoken directions, but if someone addresses me to ask me something or warn me that I'm about to do something risky, I still hear them. It is a very subjective thing, and plenty of people will not want that. Still, I get benefits from them.