Of course, you are right. I recall Geyer at some point writing some people thought there was a difference between having a burn-in period that happened to have X_b = x_b, and simply starting with X_0 = x_b. I can't find that on this page though.
I see, you're right that those two are entirely equivalent (it's what makes the Markov chain 'Markov'). And using an approximate sample instead of burn-in isn't entirely without merit, though I think I've only seen that being used with things like Brownian motion where the trajectory is the thing of interest, rather than the samples themselves.
Yes, he writes "If you start at x and I start at x then your MCMC run is no better than mine." and surrounding text is making this point.