Isn't the distribution independent of the time the arm $i$ was chosen?
Each one of the two references you describe assumes the context of the random bandit problem proposed by Robbins (1952) where the underlying reward distributions of each bandit are fixed. Therefore, yes, the underlying distributions are independent of the current time.
Is it because the chosen arm at step $t$ (namely $I_t$) is a random variable and $X$ depends on it?
The reward is a random variable which is dependent on the chosen arm at time $t$. Since each arm has an underlying reward distribution, the index $I_t$ is a random variable that designates the specific arm we are pulling, and the index $t$ denotes the time step when we pull the arm.
Why is $t$ used twice in the index (namely $I_t,t$)?
Note that $t$ is used twice, but the observed value of $I_t$ does not encode any information about the time it was chosen. For example, if $I_m = 5$, then $X_{I_m,\ m} = X_{5,\ m}$. If we drop the second subscript, then we have no way to distinguish $X_{5,\ m}$ notationally from $X_{5,\ n}$ (where $I_{n\ \neq\ m} = 5$). Two distinct rewards $X_{5,\ m}$ and $X_{5,\ n}$ would map to the same reward $X_5$ notationally. At first glance, this introduces numerous potential notational problems, such as losing the count of the number of times each arm was pulled.
Why not simply use $X_i$ instead of $X_{I_t,\ t}$ (in terms of rewards)? Shouldn't $X_{I_t}$ be sufficient, since $X_{I_t,\ m}$ and $X_{I_t,\ n}$ are drawn from the same distribution?
Admittedly, there are probably ways to get around the extra subscript for certain algorithms. For example, maybe you are using an algorithm where past rewards from each arm are averaged to yield an estimate of each arm's expected reward (see Section 2.2 of Sutton and Barto). This may require a collection of lists that store the past rewards for each arm, or it may require the count of each arm being pulled and an associated current estimate of the expected reward (see Section 2.4 of Sutton and Barto). However, these methods introduce more parameters that would be unnecessary had we initially included a second subscript for time in our notation (e.g. the counts of each arm pulled, the current estimate of expected reward for each arm, the labels of each reward list corresponding to an arm, etc.). Most of the fundamental equations regarding multi-armed bandits that I have seen are either heavily or solely dependent on the reward random variable (e.g. the definition of regret). Keeping the time index in a single random variable promotes concision and consistency among various sources by preventing the need for delegating the time index to another random variable, data structure, etc., even though specific implementations or contexts may profit from other notations.
The dual-subscript notation also has the benefit of generalizing to other contexts aside from the one posed by Robbins (1952). These include nonstationary reward distributions (see Section 2.5 of Sutton and Barto), time discounting, and families of alternative bandit processes, among others (see Sections 2.2-2.4 of this book for info on the last two extensions).
