I don't know, of course, because the evidence at hand is based on my experience. But, I'll leave the reader to consider whether these observations generalize.
Proponents of Bayesian statistical inference argue that Bayesian credible intervals are more intuitive than the frequentist confidence intervals, because the Bayesian inference is a probability statement about a parameter. A frequentist uses the term 'confidence' because their intervals are based on the probability that a random interval includes the value of a parameter.
A non-statistician scientist might initially agree that the Bayesian interval is more natural because it reads as a probability statement. But, while scientists do often think and behave (perhaps subconsciously) in a Bayesian fashion (i.e., update their prior beliefs using evidence from current experiments), their conscious notion of probability is often more aligned with the frequentist (roughly, the relative frequency of an event in repeated experiments).
For each type of interval, the 'parameter' most often refers to a fixed population quantity used to index a family of parametric models. Hence, when the scientist is reminded of this, the conflict is apparent between the Bayesian interval interpretation and the frequentist notion of probability. The scientist may ask: 'Since I have assumed that a parameter has fixed value, how can I claim that its value will lie in an interval with some frequency in repeated experiments?'
Of course, the above is a misinterpretation (as was pointed out in a previous discussion), because the Bayesian idea of probability is different than the frequentist idea. The scientist may later learn that for Bayesians, 'probability' refers to ones subjective belief about the fixed value of a parameter, but that this belief may be modified by new evidence. The fully informed scientist, despite any subconscious Bayesian tendencies, will often reject the Bayesian notion of probability in favor of the more 'objective' frequentist probability.
So, how should a Bayesian argue more convincingly?
I suppose the title of this post might have been "Bayesian vs. Frequentist Probabilities: ..." I would be surprised if there were not a literature related to this question that I have neglected. But, isn't a blog a reasonable forum to express ideas without the need for exhaustive research to ensure someone else hasn't had the same idea?