Does $\langle x_i\rangle_{i\in I}$ refer to a function $x$ or to the image of $x$ ($x[I]$)?

Question

I am confused with the notation and jargon for families.

Wikipedia (https://en.wikipedia.org/wiki/Indexed_family) says "indexed families and mathematical functions are technically identical ..." and goes on to use the notation $(x_i)_{i\in I}$ to refer to such "families".

Proof Wiki (https://proofwiki.org/wiki/Definition:Indexing_Set/Family) says "Let $\langle x_i\rangle_{i\in I}$ denote the set of the images of all element $i\in I$ under $x$." It further goes on to say that $\langle x_i\rangle_{i\in I}$, $(x_i)_{i\in I}$, and $\{x_i\}_{i\in I}$ all denote the same set.

In Set Theory and Logic by Robert R. Stoll it says "Suppose that $y$ is a function ... the function $y$ itself [is called] a family ... it is commonplace to write "$\{y_i\}$ with $i\in I$".

It seems that the notations $\langle x_i\rangle_{i\in I}$, $(x_i)_{i\in I}$, and $\{x_i\}_{i\in I}$ are all the same but refer variously to either a function or the image of a function.

So it seems that the options are, for some function $x:I\to\text{im}(I)$:

1. $\langle x_i\rangle_{i\in I}=x$
2. $\langle x_i\rangle_{i\in I}=\text{im}(I)$

All I seem to know right now is that for some $f:X\to Y$
$f$ is a "family" or "indexing function"
$X$ is an "index set" or "indexing set"
$f(i)$, notated $f_i$ is the "$i$th coordinate"