Mathematical description of a random sample

Let's say that the result of an experiment is a n-tuple of real numbers. When we accept 1. as a model of our experiment, we have a probability space $\Omega$ and a random variable $$ X: \Omega \to \mathbb{R}^n $$ The outcome of an experiment corresponds to a $\omega \in \Omega$ and therefore to an n-tuple $(X_1(\omega), ..., X_n(\omega))$. This model allows us to ask if the elements of this n-tuple are independent and if not, what their joint distribution is.

If we accept 2. as a model, we have a probability space $\Omega$ and a tuple of random variables $$ X_i: \Omega \to \mathbb{R} $$ so that the n-tuple is a random variable of the probability space $\Omega^n$ (Cartesian product). So, in this case, the independence of the elements of the tuple is built into the model. If the elements of the tuple are supposed to be independent, it does not matter.

Note that in the first case we can set $X_i = X_j$, either strict or modulo a null set; in this case we will have a tuple of identically distributed random variables. Choice no.1 does not necessarily imply that the elements of the n-tuple are different random variables (either strictly different or modulo a null set).

Thanks to the stimulating discussion with @Didier, I've clarified something for myself. From the technical standpoint, we have $n$ random variables in the option 1, and $n$ numbers in the option 2. The problem with this may be illustrated by the following example. Consider 3 different people producing their own random samples of size 5 by throwing a die. Here is what they get:

1 person: 1, 3, 1, 4, 2 $\;\;\;\rightarrow$ $X_1(\omega'), X_2(\omega'), ..., X_5(\omega')$
2 person: 2, 2, 1, 6, 3 $\;\;\;\rightarrow$ $X_1(\omega''), X_2(\omega''), ..., X_5(\omega'')$
3 person: 1, 4, 2, 1, 3 $\;\;\;\rightarrow$ $X_1(\omega'''), X_2(\omega'''), ..., X_5(\omega''')$

On the right side, I used option 1 to code these outcomes. How to code them using option 2? We can try this:

1 person: 1, 3, 1, 4, 2 $\;\;\;\rightarrow$ $X(t_1), X(t_2), ..., X(t_5)$
2 person: 2, 2, 1, 6, 3 $\;\;\;\rightarrow$ $X(t_1), X(t_2), ..., X(t_5)$
3 person: 1, 4, 2, 1, 3 $\;\;\;\rightarrow$ $X(t_1), X(t_2), ..., X(t_5)$

($t$ is for "trial number"). Well, this clearly doesn't work. What about this:

1 person: 1, 3, 1, 4, 2 $\;\;\;\rightarrow$ $X(\omega_1), X(\omega_3), ..., X(\omega_2)$
2 person: 2, 2, 1, 6, 3 $\;\;\;\rightarrow$ $X(\omega_2), X(\omega_2), ..., X(\omega_3)$
3 person: 1, 4, 2, 1, 3 $\;\;\;\rightarrow$ $X(\omega_1), X(\omega_4), ..., X(\omega_3)$

This seems to work, but how to write it in a general manner?

person ?: ?, ?, ?, ?, ? $\;\;\;\rightarrow$ $X(?), X(?), X(?), X(?), X(?)$

I think it is this point at which we arrive to an appropriate general notation specified in the option 1.