> So does the Wikipedia entry need to be corrected?

I don't think so. I think there are two definitions of reversibility
going on here. The first one, this is the one we've already talked about.

I think the first is just a fancy way of saying that going from $i$
to $j$ occurs with the same probability as going from $j$ to $i$, in the
reversed process.

Regarding the second, I think it's talking about the
long-time stationary behavior. (I didn't see the book but in other places I've seen them
talking about a discrete version of this).

In other words, when does

$$ |\pi \rangle = \lim_{t\to \infty} \exp(t L) | \hat 1 \rangle =
\lim_{t \to \infty} \exp(t \widetilde L) | \hat 1 \rangle $$

So when does the reversed process have the same steady state as the forward process? In other words, when does changing the directions of the arrows change the stationary state.