Is this how a Bayesian bootstrap works?

I am a bit new to the whole nonparametric and Bayesian idea, so tell me if this is correct: to estimate, say, the mean of a dataset's population we do the following:

1. We define a function $f(x)$ that is the PDF of our prior assumption of the distribution of the population.
2. We use MCMC to resample independently from our dataset under probability $f(x)$ for each sample.
3. After there are a lot of bootstrap samples, evaluate the statistic (mean) on each.
4. The CI is the two quantiles of the approximated bootstrap distribution that match our confidence level.

I read a little bit about reweighting, etc. but I'm confused. Do we use the prior distribution to reweight the samples (e.g. by Metropolis-Hastings sampling) or to reweight the statistic calculated on individual resamplings? Or both?

After all, the distribution of our confidence in the statistic is different from the distribution of individuals in the population.