I have a question about the distribution of a pivot. Different sources give slightly different definitions of a pivot. Some of them define it as a random variable $Q(\mathbf{X}, \theta)$ whose distribution is the same for all $\theta$, where $\theta$ is a vector of parameters of $X$'s distribution (see definitions from wikipedia and Casella-Berger below; here $X$ is a r.v. which generate observations). But some other sources define pivot as a random variable whose distribution doesn't depend on unknown parameters of $X$'s distribution (see definition from "probability course" below), this means that theoretically pivot's distribution can depend on known parameters of $X$'s distribution. All examples of pivots for different population distributions, that I have seen in textbooks, didn't depend on all (known and unknown) parameters of $X$'s distribution.
So, my question is – can the distribution of a pivot depend on known parameters of $X$'s distributiion?
I mean cases when $X \sim P_\theta; ~\theta = (\theta_1, \ldots, \theta_k),$ and some of the components of vector $\theta$ are known (known parameters) and other components are unknown (unknown parameters); and we need to build confidence interval for some unknown parameter $\theta_j$ using sample from $P_\theta$ (typically we construct a pivot for this purpose).
Below definitions of a pivot from three different sources are given.
First one is the definition of a pivot from wikipedia:
In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters). A pivot quantity need not be a statistic — the function and its value can depend on the parameters of the model, but its distribution must not. If it is a statistic, then it is known as an ancillary statistic.
More formally, let $\mathbf{X} = (X_1,X_2,\ldots,X_n)$ be a random sample from a distribution that depends on a parameter (or vector of parameters) $\theta$. Let $g(\mathbf{X},\theta)$ be a random variable whose distribution is the same for all $\theta$. Then $g$ is called a pivotal quantity (or simply a pivot).
And here is the definition of a pivot from Casella-Berger's book "Statistical inference":
Finally, here is the definition of a pivot from the website probabilitycourse.com:
