Barking orders at somebody is to lead by authority. Leading with your team is to lead by influence.
Laughing at somebody is to mock them. Laughing with somebody is to create or share a good moment.
Teaching at a class is to enforce obedience. Teaching with somebody is to learn and solve interesting problems together.
The difference between “at” and “with” is enrollment. When enrollment is absent, authority is forced to fill the vacuum.
Inspiration: Seth Godin’s work
Observations 