I haven't closely followed your discussion and just briefly glanced into the book but p.23 Theorem 1.8 says that if a Markov process/chain is STATIONARY then the above holds.
every reversible markov process seems to be stationary (Lemma 1.1) but the reverse is not true but only iff it is
detailed balanced (theorem 1.2).
That is if its reversible the transition probabilities satisfy , but if  holds then in order
for the theorem to apply it seems the Markov process has to be stationary in order to yield reversibility. In order for the
detailed balanced condition to be satisfied the "summation has to be finite" (see proof), which seems to be guaranteed
only if one assumes that the chain is stationary (I don't see this at the moment but I don't have infinite time to delve into this). So does the Wikipedia entry need to be corrected?