How do Bayesians interpret $P(X=x|\theta=c)$, and does this pose a challenge when interpreting the posterior? I have seen the post Bayesian vs frequentist interpretations of probability and others like it but this does not address the question I am posing.  These other posts provide interpretations related to prior and posterior probabilities, $\pi(\theta)$ and $\pi(\theta|\boldsymbol{x})$, not $P(X=x|\theta=c)$.  I am not interested in the likelihood as a function of the parameter and the observed data, I am interested in the interpretation of the probability distribution of unrealized data points.
For example, let $X_1,...,X_n\sim Bernoulli(\theta)$ be the result of $n$ coin tosses and $\theta\sim Beta(a,b)$ so that $\pi(\theta|\boldsymbol{x})$ is the pdf of a $Beta(a+\sum x,b + n - \sum x)$.
How do Bayesians interpret $\theta=c$?  $\theta$ of course is treated as an unrealized or unobservable realization of a random variable, but that still does not define or interpret the probability of heads. $\pi(\theta)$ is typically considered as the prior belief of the experimenter regarding $\theta$, but what is $\theta=c$?  That is, how do we interpret a single value in the support of $\pi(\theta)$?  Is it a long-run probability?  Is it a belief?    How does this influence our interpretation of the prior and posterior?
For instance, if $\theta=c$ and equivalently $P(X=1|\theta=c)=c$ is my belief that the coin will land heads, then $\pi(\theta)$ is my belief about my belief, and in some sense so too is the prior predictive distribution $P(X=1)=\int\theta\pi(\theta)d\theta=\frac{a}{a+b}$.  To say "if $\theta=c$ is known" is to say that I know my own beliefs.  To say "if $\theta$ is unknown" is to say I only have a belief about my beliefs.  How do we justify interpreting beliefs about beliefs as applicable to the coin under investigation?
If $\theta=c$ and equivalently $P(X=1|\theta=c)=c$ is the unknown fixed true long-run probability for the coin under investigation: How do we justify blending two interpretations of probability in Bayes theorem as if they are equivalent?  How does Bayes theorem not imply there is only one type of probability?  How are we able to apply posterior probability statements to the unknown fixed true $\theta=c$ under investigation?
The answer must address these specific questions.  While references are much appreciated, the answers to these questions must be provided.  I have provided four Options or proposals in my own solution below as an answer, with the challenges of interpreting $P(X=x|\theta=c)$ as a belief or as a long-run frequency.  Please identify which Option in my answer most closely maps to your answer, and provide suggestions for improving my answer.
I am not writing $P(X=x|\theta=c)$ to be contemptuous.  I am writing it to be explicit since $P(X=x|Y=y)$ is not the same thing as $P(X=x|Y)$.  One might instead be inclined to write in terms of a sample from the prior and use an index of realizations of $\theta$.  However, I do not want to present this in terms of a finite sample from the prior.
More generally, how do Bayesians interpret $P(X=x|\theta=c)$ or $P(X\le x|\theta=c)$ for any probability model and does this interpretation pose any challenges when interpreting $P(\theta=s|\boldsymbol{x})$ or $P(\theta\le s|\boldsymbol{x})$?
I've seen a few other posts tackle questions about Bayesian posterior probability, but the solutions aren't very satisfying and usually only consider a superficial interpretation, e.g. coherent representations of information.
Related threads:
Examples of Bayesian and frequentist approaches giving different results
Bayesian vs frequentist interpretations of probability
UPDATE:
I received several answers.  It appears that a belief interpretation for $P(X=x|\theta=c)$ is the most appropriate under the Bayesian paradigm, with $\theta$ as the limiting proportion of heads (which is not a probability) and $\pi(\theta)$ representing belief about $\theta$.  I have amended Option 1 in my answer to accurately reflect two different belief interpretations for $P(X=x|\theta=c)$.  I have also suggested how Bayes theorem can produce reasonable point and interval estimates for $\theta$ despite these shortcoming regarding interpretation.

Below are four different interpretations using the coin toss example that was provided in the original question.  Option 1 a) appears to be the appropriate interpretation under the Bayesian paradigm.  If you find one of these that maps to your answer, please identify it and offer suggestions for improvement if needed.
Option 1: Probability statements about $X$ and probability statements about $\theta$ are both statements of personal belief.
a) $\theta$ is the limiting proportion of heads as the number of flips tends to infinity for the coin under investigation, an unknown fixed constant with value $c$ and is not a probability.  The only valid probability is my belief about any given flip, which I set equal to this unknown fixed constant.  Therefore, $P\{X=1|\theta=c\}=c$ is my personal belief about the coin landing heads in any given throw if I know this limiting proportion.  Since I do not know what to believe, $\pi(\theta)$ is my belief about the limiting proportion (not my belief about the probability of heads).  If I were to integrate the data pmf using the prior distribution I would get the prior predictive distribution.  Then $P\{X=1\}=\frac{a}{a+b}$ where $\frac{a}{a+b}$ is a "known" constant.  In a different sense this would be my belief about the coin landing heads when not knowing the limiting proportion.  The posterior is my belief about the limiting proportion given the observed data.  Nevertheless, the prior and posterior probabilities do not represent factual statements about the limiting proportion of heads for the coin under investigation, nor are they statements about the experiment.  Option 1 a) amounts to Option 2 b) since this is how the posterior is operationalized in practice using Monte Carlo simulations.
b) $c$ is the limiting proportion of heads as the number of flips tends to infinity for the coin under investigation, an unknown fixed constant and not a probability.  The only valid probability is my belief about any given flip, which I set equal to this unknown fixed constant.  Therefore $\theta=c$, and equivalently $P\{X=1|\theta=c\}=c$, is my personal belief about the coin landing heads in any given throw if I know this limiting proportion.  Since I do not know what to believe, $\pi(\theta)$ is my belief… about my belief.  If I were to integrate the data pmf using the prior distribution I would get the prior predictive distribution.  Then $P\{X=1\}=\frac{a}{a+b}$ where $\frac{a}{a+b}$ is a "known" constant.  In a different sense this would also be my belief about my belief.  This option has us applying a belief probability measure to a belief probability measure.  Similarly for $\pi(\theta|\boldsymbol{x})$ and $P\{X=1|\boldsymbol{x}\}$.  However, because of my original correspondence we can interpret prior and posterior probabilities as beliefs about the limiting proportion of heads for the coin under investigation.  Nevertheless, these prior and posterior probabilities do not represent factual statements about the limiting proportion of heads for the coin under investigation, nor are they statements about the experiment.  Option 1 b) amounts to Option 2 b) since this is how the posterior is operationalized in practice using Monte Carlo simulations.
Option 2: Probability statements about $X$ and probability statements about $\theta$ both have a frequentist interpretation.
a) $\theta=c$, and equivalently $P\{X=1|\theta=c\}=c$, is the limiting proportion of heads as the number of flips tends to infinity for the coin under investigation, an unknown fixed constant.  The density $\pi(\theta)$ depicts a collection of $\theta$’s (coins) or the limiting proportions of randomly selected $\theta$’s (coins) as the number of draws from $\pi(\theta)$ tends to infinity.  These probabilities are considered known constants.  The unknown true $\theta=c$ under investigation was randomly selected from the known collection or prevalence of $\theta$’s according to $\pi(\theta)$, and the observed data is used to subset this collection forming the posterior.  If we are to apply these posterior probability statements to make inference on the unknown true $\theta$ (coin) under investigation we have to change our sampling frame.  We must imagine instead that the unknown true $\theta$ was instead randomly selected from the posterior.  This, then, has cause and effect reversed since the posterior distribution, from which we selected $\theta$, depends on the data… but the data depended on the $\theta$ we had not yet selected from the posterior.  We could imagine drawing a new $\theta$ (coin) from the posterior, but this would not be the same $\theta=c$ we started with under investigation.  The challenge here is applying the probability statement in the posterior distribution to the unknown true $\theta$ (coin) under investigation in a meaningful way.
b) $\theta=c$, and equivalently $P\{X=1|\theta=c\}=c$, is the limiting proportion of heads as the number of flips tends to infinity for the coin under investigation, an unknown fixed constant.  The density $\pi(\theta)$ depicts a collection of other $\theta$’s (coins) I have given to myself or the limiting proportions of randomly selected $\theta$’s (coins) as the number of draws from $\pi(\theta)$ tends to infinity.  These probabilities are considered known constants.  The observed data is used to subset this collection forming the posterior.  The posterior is a legitimate sampling distribution of $\theta$'s (coins) I have given myself.  The challenge here is applying the probability statements in the posterior distribution to the unknown true $\theta$ (coin) under investigation in a meaningful way since at no point was the true $\theta$ (coin) sampled from the posterior.
Option 3: Probability statements about $X$ have a frequentist interpretation and probability statements about $\theta$ represent personal belief.
$\theta=c$, and equivalently $P\{X=1|\theta=c\}=c$, is the limiting proportion of heads as the number of flips tends to infinity for the coin under investigation.  Since I do not know this limiting proportion, $\pi(\theta)$ is my personal belief about this unknown fixed quantity.  On the surface this seems the most reasonable.  However, this would have Bayes theorem blending two different interpretations of probability as if they are compatible or equivalent, and it does not provide a clear link between posterior probability and the unknown fixed true $\theta$ under investigation.  This would mean we are dealing with Option 1 or Option 2.  Even if one insists on two different yet compatible interpretations of probability, Option 3 amounts to Option 2 b) since this is how the posterior is operationalized in practice using Monte Carlo simulations.
Option 4: Probability statements about $X$ have a frequentist interpretation and there are no probability statements about $\theta$.
$\theta=c$, and equivalently $P\{X=1|\theta=c\}=c$, is the limiting proportion of heads for the coin under investigation.  The reason Bayesian statistics can provide reasonable point and interval estimates despite the shortcomings above regarding interpretation is that at the core of every prior is a likelihood.  Something was witnessed or observed that gave rise to a likelihood, and therefore the prior.  There are in fact no probability statements about $\theta$, belief, long-run, or otherwise.  Bayes theorem amounts to multiplying independent likelihoods equivalent to a fixed effect meta-analysis, except the Bayesian normalizes the joint likelihood instead of inverting a hypothesis test.  If we view the Bayesian inference machine as a frequentist meta-analytic testing procedure the shortcomings above vanish.  The posterior is an asymptotic confidence distribution.  Bayesian belief is more objectively viewed as confidence based on frequency probability of the experiment.
