Background:
A stochastic process is typically defined as a collection of random variables indexed by some set, $$\{X(t): t \in T \} $$
For simplicity, assume that the indexing set is $\mathbb{N}$ and that the random variables are real-valued, i.e. measurable maps $\Omega \to \mathbb{R}$, where $\Omega$ is the sample space of a probability space but understood implicitly to represent the entire triple which is the probability space.
The definition of stochastic processes leads us to interpret them as being elements of the set $(\mathbb{R}^{\Omega})^{\mathbb{N}} $, where $\mathbb{R}^{\Omega}$ denotes the space of all real-valued random variables, and analogously $(\mathbb{R}^{\Omega})^{\mathbb{N}}$ denotes the set of all (suitably measurable) functions $\mathbb{N} \to \mathbb{R}^{\Omega}$.
However, a stochastic process can also be identified as a random variable mapping from the probability space $\Omega$ to the space of functions $\mathbb{N} \to \mathbb{R}$, denoted $\mathbb{R}^{\mathbb{N}}$, or in other words, as an element of the set $(\mathbb{R}^{\mathbb{N}})^{\Omega}$, denoting measurable functions $\Omega \to \mathbb{R}^{\mathbb{N}}$. (See also.)
Now of course, using the properties of exponential objects, and since $\Omega \times \mathbb{N} \cong \mathbb{N} \times \Omega$, one has that $$(\mathbb{R}^{\Omega})^{\mathbb{N}} \cong \mathbb{R}^{\Omega \times \mathbb{N}} \cong \mathbb{R}^{\mathbb{N} \times \Omega} \cong (\mathbb{R}^{\mathbb{N}})^{\Omega} $$ and thus the two definitions of stochastic processes are equivalent.
Question: Do these two different interpretations of stochastic processes have names or terminology associated with them (in the mathematical literature)?
Motivation:
From the point of view of a physicist, using a stochastic process to model the random movement of a particle over time, the first definition/interpretation allows one to ask questions about the particle at a given time, but not for a given trajectory -- e.g., what is the probability that at time $N$ the particle will be within this region? In contrast, the second interpretation/definition doesn't allow one to ask such questions, but forces one to ask questions in terms of trajectories -- e.g., what is the probability that the particle's trajectory will cross the $x$-axis three or more times before time $N$?
Since the two points of view are isomorphic, the distinction may seem meaningless to a mathematician. For example, the course I took about stochastic processes in a mathematics department switched freely between the two interpretations/definitions, with the professor not even bothering to note (or consciously realize) that one could construe the two as different. But when I took another course about stochastic processes in a physics department, when I asked the professor questions about processes from the second (trajectory) viewpoint, he had to restrain himself from laughing (because to him it seemed so unnatural or unnecessarily complicated to ever think about stochastic processes that way).
So I imagine that there aren't any sources in the mathematical literature which bother to make any distinction between the two points of view, much less assign to either of them formal names or definitions, but I figured I might as well ask in any case.
Here are two related questions which also note the distinction between these two different points of view regarding stochastic processes: (1)(2).