Is a Nash equilibrium anything more than what it is?
The conventional definition of Bayesian Nash equilibrium (BNE) is as follows:


  A pure strategy BNE is a profile of
  type-contingent strategies $$(s_i(\theta_i),s_{-i}(\theta_{-i}))=(s_1(\theta_1),\dots,s_{i-1}(\theta_{i-1}),s_i(\theta_i),s_{i+1}(\theta_{i+1}),\dots,s_m(\theta_m))$$
  such that for each $i\in P$,  $$ s_i(\theta_i)\in\arg\max_{s_i'\in
 S_i} \sum_{\theta_{-i}}p(\theta_{-i}\mid \theta_i) 
 u_i(s_i'(\theta_i),s_{-i}(\theta_{-i}),(\theta_i,\theta_{-i})). $$


If we want to define BNE with the language of beliefs about the other player's actions, we may propose the following definition: 


  A pure strategy BNE is a profile of type contingent strategies
  $(s_i(\theta_i),s_{-i}(\theta_{-i}))$ and a profile of beliefs
  $b^*=\{b_i(\theta_{-i})\}_{i\in P}$, where $b_i(\cdot)$ is a
  distribution over $S_{-i}(\cdot)$ conditional on $\theta_{-i}$, such
  that 
  
  
  $s_i(\theta_i)$ maximizes $i$'s expected utility (expectation over $\theta_{-i}$) given $b_i(\cdot)$ and a prior on types; and 
  $b_i(\cdot)=s_{-i}(\cdot)$, i.e. beliefs in $b^*$ are correct.
  


Thus if the type distribution is degenerate, the definitions of BNE and NE coincide. Nevertheless, the correctness of beliefs (about the opponents' strategies) are present in both definitions. 
The difference is that in BNE, players' beliefs are over a set of functions (due to the fact that in Bayesian games, strategies are type-contingent).