Sorry, when I said ‘only the Login or Logout menu items show’ I thought it would be apparent that I meant that the Register link doesn’t appear on the menu whether I’m logged out or in.
So, to be clear, I have added ‘Login’ and ‘Register’ to the default menu from the ‘Logged out’ Buddypress section, and ‘Logout’ to the default menu from the ‘Logged in’ Buddypress section.
When I am logged in ‘Logout’ shows in the menu.
When I am logged out ‘Register’ does not appear in the menu, only ‘Login’ does.
As they were both added to the same menu from the same section of the ‘Add to menu’ options this can’t be expected behaviour surely.