Bayes-Nash equilibrium and correctness of beliefs Define a Bayesian game as follows: $$G = \left\langle I, \left(A_i,T_i,(p_{t_i})_{t_i \in T_i}, u_i \right)_{i \in I} \right\rangle$$


$I$ is the set of players 
$A_i$ is the action set for player $i$, 
$T_i$ is the set of possible types for player $i$, 
$p_{t_i} \in \Delta(T_{-i})$ is player $i$'s beliefs regarding the types of the other players. $(T_{-i}=\times_{j \ne i}T_j)$
$u_i : A \times T \rightarrow \mathbb{R}$ is player $i$'s utility function


Then a Bayes-Nash equilibrium is defined as follows:


  A (pure) Bayes-Nash equilibrium is a profile of choice functions (or
  strategies) $(\sigma_i:T_i \rightarrow A_i)_{i \in I}$ such that,
  $\forall i \in I, \forall t_i \in T_i, \forall a_i \in A_i$,
  
  $$\sum_{t_-{i}}p_{t_i}(t_{-i}) \cdot u_i(\sigma_i(t_i),
  \sigma_{-i}(t_{-i}); t_i, t_{-i}) \geq \sum_{t_-{i}}p_{t_i}(t_{-i})
  \cdot u_i(a_i, \sigma_{-i}(t_{-i}); t_i, t_{-i})  $$
  where, for every $t_{-i}, \sigma_{-i}(t_{-i}) = (\sigma_j(t_j))_{j\ne i}$.


And now my question: Am I correct that this implies that, in a BNE, every player (or rather, every type of every player) best responds given their beliefs about the types of other players, but that there is nothing in a BNE that pins down the beliefs a player (or type of player) has on the types of other players? That is to say, in a BNE, a player (or type of player) could have a degenerate belief (putting full probability on some $t_j^*$ for player $j$) in an equilibrium in which $t_j \ne t_j^*$? Put more simply, can a type of player's beliefs about the type of another player be wrong in a BNE?
I think your definition is incorrect, or at least incomplete.** Usually, in a Bayesian game, there is assumed to be a prior distribution on $T$ (where $T = \times_i T_i$). This distribution is called the "common prior" and it is assumed to be common knowledge that types are drawn according to this distribution. In this case, each player $i$'s belief $p_i$ is given by Bayesian updating on $T_i$ and this prior; and in BNE player $i$ must be best-responding to this belief.

The assumption of a common prior and Bayesian update rule (which is what gives this solution concept its name, after all) mean that players cannot be wrong, merely underinformed. In other words, a player with posterior belief $p_i$ is correct about the distribution of types of other players conditioned on her own, even though she does not know which realizations they have.

** Edit. Osborne and Rubenstein's text mentions that it is possible to define a more general game in which each player has a different prior distribution (there is no common prior). So your definition does match their most general definition. I suppose that in such a case two players may hold incompatible views, hence you could say that someone must be incorrect. That all being said, the vast majority of Bayesian games assume a common prior.