Halmos gives the following definition for the Cartesian Product of an Indexed Family in Naive Set Theory, Chapter 9:
If $\{X_i\}$ is a family of sets $(i \in I)$, the Cartesian product of
the family is, by definition, the set of all families $\{x_i\}$ with
$x_i \in X_i$ for each $i \in I$.
Meanwhile, Mendelson provides the following definition in Introduction to Topology:
DEFINITION Let $\{X_\alpha\}_{\alpha \in I}$ be an indexed family of sets. The product of the sets $\{X_\alpha\}_{\alpha \in I}$,
written $\prod_{x\in I}X_\alpha$ consists of all functions $x$ with domain the indexing set $I$ having the property that for each $\alpha \in I$, $x(\alpha)\in X_\alpha$.
Here's is my attempt at parsing this as per Mendelson's definition:
Let's suppose my index set $I = \{0, 1\}$, and that $X = \{\{Alpha, Beta\}, \{Gamma, Delta\}\}$
If, for example, I say that $\{Alpha, Beta\}$ is $X_0$, and $\{Gamma, Delta\}$ is $X_1$, then the functions that could arise from this are $\{ (0, Alpha ), (1, Gamma) \}, \{(0, Beta), (1, Gamma)\}, \{(0, Alpha), (1, Delta)\},$ and $ \{(0, Beta), (1, Delta)\}$
However, there is also a version of this family that indexes $\{Alpha, Beta\}$ by $X_1$ and $\{Gamma, Delta\}$ by $X_0$, one that indexes $\{Alpha, Beta\}$ by both $X_0$ and $X_1$ (since it was never stated that this had to be a one-to-one function), and one that does the same for $\{Gamma, Delta\}$. So, if that's valid, the Cartesian Product would end up being a set containing all the resultant unordered pairs of ordered pairs, similar to those described above.
My parsing of the definition seems wrong, in large part because it's not clear to me how this aligns with Halmos's definition. I'm not clear I've understood Mendelson, for that matter, but at least I have some notion of how to start. Most of the questions asked on this subject previously use notation I'm not at all familiar with, and I am using this simple example because it seems more likely I'll understand the inevitable corrections in this context.
Best Answer
Mendelson's definition is correct: products are sets of functions. Halmos' is too: an indexed family $\{x_i\}$ as he writes it, is also a function with domain $I$, though many don't realise it. Both are really the same notion in a slightly different notation. Personally I like to make the points explicit functions, it often makes for nicer proofs.
In your example elements are of the form e.g. $\{(0,\alpha),(1,\gamma)\}$ (a function is a set of ordered pairs, of course), which is usually just identified with $(\alpha,\gamma)$ in the finite index set case. The function view is the right view for all products, the tuples are fine for finite products.