So, the two examples at the top are not zeros:
{ 6 | -10 || 8 | -8 } { 2 | 0 || 0 | -2 }
In fact, they are not even numbers. In both cases, both options aren't numbers because the left values are bigger than the right. And hence, the entire value isn't a number, only a game.
These games aren't equal to { | } at all. When summed with other games, they will in fact have a bearing on play.
Note: The method of rendering the Go positions presented as games only applies if you are playing the sort of mathematically pure form of Go where you can pass. Hence, I think the authors confusion that these positions, in a real Go game, net no points, and hence seem to be like a "zero", whereas in a game where you are forced to play someone, Left or Right will eventually have to play.
{ 6 | -10 || 8 | -8 } { 2 | 0 || 0 | -2 }
In CGT, zero is defined as a second player win. That is true of both of these games. In the first, if Black plays first White replies to a score of -10, so White (the second player) wins; if White plays first Black replies to a score of 8 and wins. In the second, whoever plays first the second player replies to 0, which is a win for them, by the definition of 0. So, yes, both are CGT zeroes.
I hope the new page zero game will help to clarify this. It's possible that I'm duplicating content that's already here somewhere (similarly for abstract game), but I didn't manage to find an explicit definition anywhere on this site.