Specifically, before the first move, we never want any buttons,
(instead, the first player will commit to a team by making a move).
After the first move, there are two cases:
1. A player is already on a team:
In this case we want only one button, labeled "Switch" to
change to the other team.
2. A player is just spectating:
This is the only case where we need to display two buttons,
one labeled "Join X" and one labeled "Join O".