 Solution

A version of a stochastic process means a stochastic process on the same probability space such that, for every fixed $t$,The exceptional null set may depend on $t$. Indistinguishability of stochastic processes means that there is one null set outside which $X_t=Y_t$ for all $t\ge 0$ simultaneously.

For an example separating the definitions, let $U$ have uniform distribution on $(0,1)$, and setFor every fixed $t$, $\mathbb{P}(U=t)=0$, so $Y$ is a version of a stochastic process with original stochastic process $X$. But for every sample outcome the stochastic processes differ at its time $t=U$. HenceThe spike path of $Y$ is not right-continuous at $U$, which explains why the next part's regularity assumption rules out this example.