When I was young, my parents taught me ethics by telling me "always act like someone is watching you, even when they aren't". The idea was that when you think people are watching, you are more likely to not do anything wrong.
It is great advice for a manager - particularly because someone is probably always watching. Your team is always watching or listening to see how you act. They take their cue from them. If you gossip, they will think it is ok to gossip and may even worry that you gossip about them. If you act differently in front of other leaders, they will think that acting disingenuously is expected.
As is often the case, the advice you got when you were younger is usually the best.