Use thumb Bb (I call it double thumb) if there is a flat in the key signature (unless there is a C flat or a lot of B naturals)

Use chromatic Bb (lever) otherwise. 1+1 is situational (like going from F to Bb in long tone 1A from the blue book).

Why do they mark the rectangular thumb key (I call that single thumb)? Because they have to mark in one of them. If you never learn thumb Bb, then you would never have to use that other thumb key.