In Osborne's An Introduction to Game Theory Nash equilibrium is described as follows (p. 21–22):
First, each player chooses her action according to the model of rational choice, given her beliefs about the other players' actions. Second, every player's belief about the other players' actions is correct.
It seems to me that this definition is not completely equivalent to the usual definition of the Nash equilibrium as a strategy profile where each player's strategy is a best response to the strategies of the others.
The usual definition says nothing about beliefs and therefore allows for the possibility that beliefs might be incorrect.
To take a trivial possibility, consider the Prisoner's Dilemma. Suppose each player believes that the other player will not confess. Since confessing is a dominant strategy each player would still confess. So the actions constitute a Nash equilibrium even though the players' beliefs are completely the opposite of the actual equilibrium actions.
Am I right in this understanding that Osborne's definition characterizes something other than Nash's equilibrium?