Central limit theorem notation Can you tell me if my understanding of the CLT is correct? Maybe it's just a matter of notation.
The classical CLT states:
Let $X_1,...,X_i,...,X_n$ be a sequence of iid random variables drawn from a distribution with mean $\mu$ and variance $\sigma^2 < \infty$.
Then, let the sample avarage be equal to $\bar{X}_n = \frac{X_1+...+X_i+...+X_n}{n}$.
The CLT says, when $n$ becomes large then $\sqrt{n}(\bar{X}_n-\mu)$ is approx. normal with mean $0$ and variance $\sigma^{2}$.
What this theorem is saying is: take a random sample from a population, then compute the sample average and keep it aside, then you keep sampling let's say for additional 100,000 times, and you get additional (possibly different, at least most of them) 100,000 sample averages. Then, if you look at the sampling distribution of all these sample averages, it would be approximately normal.
Now, the generic $X_i$ is the sample average obtained from the $i$ random sampling or is it simply a random observation obtained from the $i$ random sampling? I mean, is it $\bar{X}_i$ or a single observation?
What I'm asking is: should I can consider $X_i$ either as a single observation (in this case each draw means to draw a single value at each trial, and the number of random draws is $n$)  or as a sample mean (and in this case I'm drawing samples fixed in size, and in this case, $n$ is the number of random samples drawn)?
Shall the same interpretation for $X_i$ hold when we present the law of large numbers?

As extensively already discussed by others and in the comments, the Central Limit Theorem tells us about the behaviour of the distribution of standardized sample averages $\bar X_i$ as $i$ becomes large, where $X_i$ and its space play no role.
In the Central Limit Theorem, the basic underlying hypothesis is

For each $n$, let $X_1, \ldots X_n$ be independent random variables.

As $n$ changes, the underlying probability space may change, but this is of no consequence since a convergence in distribution statement is a statement about convergence of a sequence of real-valued functions on $\mathbb{R}$.
Indeed, If $X_1,\ldots,X_n$ are independent with distribution functions $F_1,\ldots,F_n$ and
$$
T_n = (S_n - E(S_n))/\text{var}(S_n),
$$
with $S_n = X_1,\ldots,X_n$, the distribution function of $T_n$ is completely determined by the $F_i$, and the validity of a statement about convergence in distribution of $T_n$ is also determined by $F_i$, regardless of the construction of the underlying probability space.
