Apple's Standard Human Interface Guidelines for the iPhone suggest
that the gray Disclosure Indicator is used to reveal a new level of
information in a hierarchy or the choices associated with the item. In
this case, it's used for the latter where you can select the type of
player (currently either human or Deep Green).
Once on this level, you'll see the blue chevron button, the Detail
Disclosure Button, that indicates that you can see – and configure –
detailed information about the item. So for a human player, you can
edit the name, and for a Deep Green player you can adjust the
strength.
This is how Apple uses these elements in its application, and this is
what I've tried to do as well.
Joachim