So, obviously that list isn't exhaustive, because there are more ways to split interactions than public/private, but in an attempt to add meaningful new outlooks:

4) Speak about your weaknesses openly when in public, and deny them in private.

Many high status individuals are much harsher, demanding, arrogant, and certain in private than in public. I think this is a result of -- when you don't know the target well -- not knowing who you will have to impress, who you have to suck up to, and who is only useful when they get you the thing you want.