Correct formulation of axiom of choice

It does make sense subject to a generous interpretation of the role of the indexing set $I$ and the indexing function $i \mapsto A_i$. It would be much better style (in my opinion) either to write it as you suggested, stating explicitly that the indexing function $i \mapsto A_i$ is part of the data or to write it without mentioning the indexing set at all: "if $U$ is a set of non-empty sets, then there is a function $f$ with domain $U$ such that $f(A) \in A$ for every $A \in U$". The two statements are equivalent (because any indexing function $i \mapsto A_i$ of the set $U$ factors through the identity function on $U$, which you can regard as a sort of "minimalist" indexing function).

[Aside: I don't think there is any general agreement that "family" means indexed family. Indeed, some authors explicitly state that they use terms like "set", "family" and "collection" as synonyms. So it's safest to say "indexed set" or "indexed family".]

There are several common ways to express the axiom of choice in set theory:

• If $M$ is a set of nonempty sets, there is a function $f$ such that $f(x) \in x$ for each $x \in M$.

• • Variant: if $M$ is a set of pairwise disjoint nonempty sets, there is a set $C$ such that $|C \cap x| = 1$ for all $x \in M$.

• If $g$ is a function and $I$ is a set such that $g(i)$ is nonempty for each $i \in I$, there is a function $h$ such that $h(i) \in g(i)$ for each $i \in I$.

The latter could be viewed as a definition in terms of indexed families; we could look at $g(I)$ as an indexed family $\{ G_i : i \in I\}$ where $G_i = g(i)$. The former is somewhat easier to state, and the variant is even easier because it does not require us to define a "function".

It is also at least somewhat common in set theory to look at each set as indexed by itself, when we want to treat a set as an indexed set. So we have $A = \{ A_a : a \in A\}$ where $A_a = a$ for $a \in A$. (Say that five times fast...)

The definition linked in the question could be rephrased formally in terms of any of the definitions I mentioned. This is usually viewed as routine, particularly because all reasonable variants are equivalent to each other over ZF set theory, so in practice it does not make too much difference which formal sentence you take to represent the axiom of choice. (In fact, Kunen's classic textbook expressed the axiom of choice as "every set can be well ordered", which is well known to be equivalent over ZF.)