In a Mixed Strategy Nash Equilibrium, each player constructs their own probability distribution over the set of their respective possible strategies.
In a Correlated Equilibrium or a Coarse Correlated Equilibrium, there is only one global probability distribution which the players must take as given, and it is defined over a game's possible outcomes rather than each player's possible strategies.
These feel like fairly different animals. Why are Mixed Strategy Nash Equilibria special cases of Correlated Equilibria and Coarse Correlated Equilibria?