Baseball seems to be a symbol of decent, "average Americans" in Twilight. Charlie and Billy enjoy watching baseball, and Charlie even seems to feel more comfortable with the idea that his daughter is going on a date with Edward Cullen when he finds out that baseball is involved. That the Cullens enjoy playing baseball shows that they're actually a fairly normal American family. Sure, they put their own spin on the game, but could we really expect them to play on a normal diamond? The baseball game in Twilight displays just how similar to humans the Cullens truly are, as well as their attempt to follow social norms, even if that means they have to "tweak" that norm a little first.