Why do we use $X_{I_t,t}$ and $v_{I_t}$ to denote the reward received and the at time step $t$ and the distribution of the chosen arm $I_t$? I'm doing some introductory research on classical (stochastic) MABs. However, I'm a little confused about the common notation (e.g. in the popular paper of Auer (2002) or Bubeck and Cesa-Bianchi (2012)).
As in the latter study, let us consider an MAB with a finite number of arms $i\in\{1,...,K\}$, where an agent choses at every timestep $t=1,...,n$ an arm $I_t$ which generates a reward $X_{I_t,t}$ according to a distribition $v_{I_t}$.
In my understanding, each arm has an inherent distribution, which is unknown to the agent. Therefore, I'm wondering why the notation $v_{I_t}$ is used instead of simply using $v_{i}$? Isn't the distribution independent of the time the arm $i$ was chosen?
Furthermore, I ask myself: Why not simply use $X_i$ instead of $X_{I_t,t}$ (in terms of rewards). Is it because the chosen arm at step $t$ (namely $I_t$) is a random variable and $X$ depends on it? If I am right, why is $t$ used twice in the index (namely $I_t,t$)?
Shouldn't $X_{I_t}$ be sufficient, since $X_{I_t,m}$ and $X_{I_t,n}$ are drawn from the same distribution?
Isn't the distribution independent of the time the arm $i$ was chosen?

Yes, but you don't know which arm was chosen at time $t$, that is what $I_t$ represents. $v_i$ would represent the $i$th arms distribution, whereas you want the distribution of the arm that was chosen at time $t$, which is $v_{I_t}$.
$X_{I_t,t}$ is used to represent the arm you chose at time $t$ and the time you chose it. Imagine if we chose the following arms at respective time steps {1,2,5,2,4}, then the reward you get at time 3 (assuming in my example time started at 1) would be $X_{5,3}$. You need this notation because every time you pull arm $i$ you get a different reward, because the reward is a random variable (unless the assumption was a deterministic reward but that would not be interesting).
It has been a while since I read the papers but I assume that the distributions of the arms are stationary, however this notation is more general and would allow for non-stationary distributions.