$\textit{The stochastic processes X and Y are called a modification or versions of each other if}$ \begin{align} P(X(t) = Y(t)) = 1 \quad \textit{for all t $\in$ I} \end{align}
$\textit{The stochastic processes X and Y are called indistinguishable}$ \begin{align} P(X(t) = Y(t) \,\,\,\textit{for all $t \in I$}) = 1 \end{align}
I am trying to come up with scenarios when the distinction between versions and indistinguishable procsses becomes relevant.
In other words, are there some situation when we cannot change to a version without messing things up? And where if we would stick to an indistinguishable change we would be in the clear?
I am also looking for a situation where the former change is possible.
Examples from applications are also of interest, not just theoretical.