In the text I'm using it says:
Let X = {$X_n : 0 \leq n \leq N$} be an irreducible Markov chain such that $X_n$ has the stationary distribution $\pi$ for all $n$. The chain is called reversible if the transition matricies of X and its time-reversal Y are the same, which is to say
$$\pi_i p_{ij} = \pi_j p_{ji}$$
A transition matrix P and a distribution $\lambda$ are in detailed balance if $\lambda_i p_{ij} = \lambda_jp{ji}$. An irreducible chain X having stationary distribution $\pi$ is called reversible in equilibrium if its transition matrix P is in detailed balance with $\pi$.
Is it because in reversible in equilibrium we are not assuming that X is non-null persisntent? Which is needed to prove the existence of the the X's time reversal Y?
In that case all reversible chains would be reversible in equilibrium, but not all reversible in equilibrium chains would be reversible. Am i correct?