Misuse of confidence set I found in a literature

I found some writings (prominently on the web) that mention a sort of confidence set (c.f. confidence interval) with the ratio of coverage at which the concerned parameter is believed to belong to the set.

Assume we have observed a sample $x_1,\ldots,x_n$ with a sample statistic $y=y(x_1,\ldots,x_n)$ . Further we are given a random set $C(Y)$ based on the statistic $Y$ to estimate a parameter $\theta$ .

Then, what does it mean by saying $C(y)$ is a p% confidence set for $\theta$ ? In other words, over which distribution the probability p, at which the $\theta$ is confidently covered by $C(y)$ , is evaluated?

Note that the parameter $\theta$ is fixed and unknown (to be estimated). Then for each observation $x$ , the derived set $C(y(x))$ determines whether $\theta\in C(y(x))$ or not, which is just an indicator taking a value of 1 or 0, as a probability.

Assuming $x$ ranges over the sample space $\Omega$ , then $\theta\in C(y(x))$ for some $x\in\Omega$ and not for the others. In this case, we can naively define (replace the integral with sum for discrete random variable $X$ )

$\tau(\theta) = P(\theta\in C(Y)\mid \theta) = \int_\Omega I_{C(y(x))}(\theta) \, f_X(x)dx$

as a measurement of “well-coveredness” of $\theta$ by $C$ , where $f_X$ is the distribution function of $X$ . This is equivalent to define $\tau$ by the conditional expectation $E_\Chi[I_{C(Y)}(\theta)\mid\theta]$ over the distribution $X$ .

The confusion comes when $y$ is derived from an observation and therefore fixed. As noted above, $P(\theta\in C(y)\mid \theta)\in\{1,0\}$ holds indeed.

This is just a memo, not to claim a sort of insight or discovery. Caveat!