Even though the underlying posterior distributions are of different type, the results will be the same if the rate of the Poisson process is sufficiently small and weakly informative and consistent priors are used.
A weakly informative prior that follows a Gamma distribution with shape parameter 1 and rate parameter 2 is a suitable choice for many problems. Working with informative prior distributions requires careful handling and a close look at the quantiles.
Credible intervals express our belief that the true underlying value is contained within the interval conditional on the conducted simulation run. Contrary to that, confidence intervals are only meaningful for a large number of repeated simulation runs.