Correlated Equilibrium

Question

I have a question about the definition of the correlated equilibrium.

I see that some authors define it as "expected payoff of playing the recommended strategy is no less than playing another strategy". Here the inequalities are for each recommended strategy of each player. (http://www.stanford.edu/~saberi/lecture6.pdf)

Others define it as "expected payoff is no less than playing other strategies". Here the inequalities are one per player. (http://ocw.mit.edu/courses/economics/14-123-microeconomic-theory-iii-spring-2010/lecture-notes/MIT14_123S10_lec08.pdf)

So which one is correct?

Are they both equivalent ?

I think that the first implies the second by the second does not imply the first.

Thanks.