Much like the definitions of "countable", "sequence" and "natural numbers" so does the definition of "family" can be changed from one context to another.
First note that the first and third interpretations are the same, they just use different letters. If we also require that the indexing is injective, that is to say that no set appears twice, then all the three interpretations coincide.
I think that the prevailing use today is generally the first/third interpretation, but you can also find the second one being used quite often. In those cases, this is actually like considering the first/third interpretation with the requirement that the indexing function is injective.
More often then not the writer expects the reader to be able and establish the proper context from the text. Not for every definition, of course, but for the very basic and implicit ones -- like the definition of a family.
To your edit, when we say that $A$ is a family, regardless to how we treat, $\bigcup A$ is the union of the sets in the family, this is sort of an abuse of notation which really eases up on notation and formality once you are aware of it.
This might be a good reason to think of a family a set, rather than a function, too.
Best Answer
I don’t like either notation: I would write $\{A_i:i\in I\}$ or $\langle A_i:i\in I\rangle$. Technically there is no difference: each implies the existence of a function $i\mapsto A_i$ whose domain is $I$. The difference is one of emphasis: when I write $\{A_i:i\in I\}$, I’m thinking of this simply as a collection of sets, whereas when I write $\langle A_i:i\in I\rangle$, I’m emphasizing the existence of the function whose domain is $I$ and whose range is that collection of sets. I might let $\mathscr{A}=\{A_i:i\in I\}$ and simply talk about the collection $\mathscr{A}$ of sets, without any reference to the specific indexing, but when I write $\langle A_i:i\in I\rangle$, the specific indexing is very much on my mind: $\langle A_i:i\in I\rangle$ is an abbreviation for a function $I\to\mathscr{A}:i\mapsto A_i$.
For a more familiar example of the distinction, compare $\{x_n:n\in\Bbb N\}$ and $\langle x_n:n\in\Bbb N\rangle$, where each $x_n\in\Bbb R$. In each case $x_n$ is just a handier notation for $\varphi(n)$, for some function $\varphi:\Bbb N\to\Bbb R$. However, when I write $\{x_n:n\in\Bbb N\}$ I’m not thinking of that function; I’m thinking of its range, the set of values that it assumes. When I write $\langle x_n:n\in\Bbb N\rangle$, however, I’m thinking of the function: this is a real-valued sequence, i.e., a function from $\Bbb N$ to $\Bbb R$, not just a countable set of real numbers.
(Note: Many people use parentheses for my angle brackets; I prefer the angle brackets for this specific notational purpose, since parentheses already have more than enough meanings.)