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

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.