It sounds like #2's daughter approaches the game as "I'd like to win Game X", rather than "I'd like to have fun playing Game X". I think putting the game away if she isn't having fun is a good idea, and then gently asking about her motivation and expectations around suggesting playing the game. Is she getting a message from somewhere that she needs to prove herself by winning?

Don’t forget the possibility of a persistent subtext of, “With your intelligence…” (not only legendary prodigies but comic book omnidisciplinary geniuses are realistic role models, and you suck if you fall short of them. Also, Smart People Play (And Win! Blindfolded against multiple opponents!) Chess/Go/Weiqi.)