You could call them Bernoulli distributions representing aleatory uncertainty on a single coin flip, I suppose. Bayesian updates of purely aleatory uncertainty aren't very interesting, though, are they? Your evidence is "I looked at it, it's heads", and your posterior is "It was heads that time".

I suppose you could add some uncertainty to the evidence; maybe we're looking at a coin flip through a blurry telescope? But in any context, Bernoulli distributions from a finite-dimensional probability distribution space mean that Bayesian updates on them are still well-posed. The concern here is that infinite-dimensional spaces of probability distributions don't always lead to well-posed Bayesian updates, depending on what metric you use to define well-posed. If there's also a concern that this can happen on Bernoulli distributions then I'd like to see an example; if not then that's a red herring.