Is the concept of a version of a stochastic process related to the concept of a version of a conditional distribution?
And is a regular version of a stochastic process somehow the same thing as the regular version of a conditional distribution?
I find this confusing because both concepts seem like they are supposed to resolve ambiguities arising when going from the countable to the uncountable case, but I rarely see authors use or explain the terminology rigorously, seemingly because they already understand it.
Note: regular version of a stochastic process refers to a version which is regular in the sense that it has regular paths (right-continuous, cadlag, continuous).
I saw a more precise definition in a textbook once, but now I can't remember which textbook it was. It was in any case definitely along the lines of the implicit definition used in other texts, namely a version of the stochastic process with regular paths.
UPDATE: I found the following definition of regular function on Wikipedia in the context of integration by parts: https://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integration#Integration_by_parts
It is very similar to the notions I mentioned earlier for functions with countably many jump discontinuities (since one can modify the value of the function at those points, with no change to its integrals, to make it regular vis-a-vis cadlag).
A function $f$ is regular at a point $a$ if and only if the left and right hand limits $f(a-)$ and $f(a+)$ both exist and the function takes the average value at $a$, i.e. $$f(a)= \frac{f(a-)+f(a+)}{2}$$
This is some extension of @jdods's comment.
A stochastic process can be understood as a family $X = \{X(t), t\in \mathbb T\}$ of random variables indexed by some parametric set $\mathbb{T}$. Another process $Y = \{X(t), t\in \mathbb T\}$ is a version of $X$ if for all $t\in \mathbb T$ $P(X(t) = Y(t)) = 1$.
What is a conditional distribution, or, more generally, a conditional probability measure (with respect to some $\sigma$-algebra $\mathcal G$)? It is defined as $\mathsf P (A \mid \mathcal G) = \mathsf E[\mathbf 1_A \mid \mathcal G]$, $A\in \mathcal B$. So this is a random set function, or, in other words, a collection of random variables indexed by the $\sigma$-algebra $\mathcal B$. So it is nothing else but a stochastic process with $\mathbb T = \mathcal B$. Moreover, a version of this stochastic process is precisely a version of conditional probability.
Concerning the regularity, there are some analogies as well. A regular conditional probability measure is a version of conditional probability which is a probability measure $\omega$-wise. So this is a version of the stochastic process $\{\mathsf P (A\mid \mathcal G), A\in \mathcal B\}$ having certain good properties: additivity, continuity.
An even stronger connection between these notions manifests if we speak of a regular conditional distribution. The latter is a version of conditional distribution $\mathsf P(X\in A\mid \mathcal G)$, which is also a probability measure on $\mathcal B(\mathbb{R})$. A regular conditional distribution is best described by its regular conditional cumulative distribution function $F_{(X\mid \mathcal G)}$. Being indexed by $\mathbb{R}$, this is a more conventional stochastic process. More precisely, a regular conditional cdf is a version of $F(t) = P(X\le t\mid \mathcal G)$, $t\in\mathbb{R}$, such that, $\omega$-wise, it is a cumulative distribution function, i.e.
$\lim_{t\to-\infty} F_{(X\mid \mathcal G)}(t)=0$, $\lim_{t\to+\infty} F_{(X\mid \mathcal G)}(t)=0$;
$F_{(X\mid \mathcal G)}$ is non-decreasing;
for each $t\in \mathbb R$, $\lim_{s\to t+}F_{(X\mid \mathcal G)}(s) = F_{(X\mid \mathcal G)}(t)$.
And it is easy to see that the regular conditional cdf $F_{(X\mid \mathcal G)}$ is just the càdlàg version of $F$. Thus, there is a strong link between regular conditional distributions and regular (= càdlàg) stochastic processes.