Say I have run a Hierarchical Bayesian model in STAN (or JAGS or BUGS) and I have the posterior samples of two slope parameters that I want to compare: $\beta_1$ and $\beta_2$. The model appears to have converged properly.

So I look at the density of the difference of the MCMC samples of the posteriors of the two slopes ($\beta_1-\beta_2$). The 95% HPD interval of this difference includes zero, and from what I've read this means something similar to a Frequentist failure-to-reject: zero is credible and hence we cannot say that $\beta_1 \ne \beta_2$.

Yet, I can see that 80% of the differences are greater than zero. Which tempts me to say that, given my assumptions and model the odds are 4-1 that $\beta_1>\beta_2$. But I don't think I've ever read anything like that.

This is so simple that it would be used everywhere if it were correct. So it must not be correct: what am I missing?