Definition of the Infinite Cartesian Product
Solution 1:
An order pair and a function $f : \{1, 2\} \rightarrow X_1 \cup X_2$ are not the same thing. Depending on your definition, some people define the order pair $(a,b)$ to be the set $\{a, \{a,b\}\}$. A function is a subset of $X_1 \times X_2$ satisfing the usual property of functions. Hence set-theoretically they are different. "Indeed, the latter concept depends on the mappings, which are defined in terms of the former one." means that functions are usually defined as a subset of $X \times Y$, i.e. subset of the set of order pairs. Hence Folland is just remarking that the definition of functions uses the notion of order pairs.
Folland only defines $(1)$ for finite cartesian product. So in the context of Folland's book, it doesn't make any sense to ask which definition is stronger for infinite families. According to folland, for infinite families there is only one notion of cartesian product, and that is the second one.
By being "prompty forgotten", is that although order pairs and certain functions from $\{1, 2\} \rightarrow X_1 \cup X_2$ are not set theoretically the same, there is a very nice bijection $\Phi$ between the two concepts. $\Phi((x_1,x_2)) = f_{(x_1,x_2)}$ where $f_{(x_1, x_2)}$ defined by $f_{(x_1,x_2)}(1) = x_1$ and $f_{(x_1,x_2)}(2) = x_2$.
(Note that its inverse would then be : for any $f : \{0,1\} \rightarrow X_1 \cup X_2$ with the property that $f(i) \in X_i$, $f$ is map to $(f(1), f(2))$. )
Solution 2:
Very briefly. Question 1 has been amply answered; the two definitions produce different sets, although there is a canonical bijection between them.
Question 2. There is no such thing as extending (1) to infinite families, it is a definition for pairs of sets only. Even if you want something for three sets, there are at least two ways to combine two Cartesian products, which give different (though isomorphic) results, neither of which involves ordered triples. But one can never get infinite products in any of these ways. One can interpret (2) as an alternative way to "generalize" this, in a manner that also caters for infinite products.
Question 3: We can forget the distinction between (1) and (2) in the case of a Cartesian product of two sets because of the canonical bijection between the sets produced by the two definitions, which means we can always and consistently translate back and forth between them when necessary. And we should forget the distinction because keeping track of which of the two is officially applied in which situation is a totally unproductive effort. The whole importance of giving these definitions is to have a precise model of a Cartesian product, so that its properties can be deduced from the axioms of set theory, but having more than one equivalent model does not add anything useful.
Absent question: if (2) can do all that (1) can (slightly differently but equivalently) why should we care about (1) in the first place? Because without (1) we would be in a chicken-and-egg situation when trying to formulate (2), not only because (2) produces sets of mappings, which require a Cartesian product (of two sets), but also because the notion of an indexed family of sets itself is defined in terms of mappings.
Solution 3:
Whey they're different
The usual set-theoretic construction of functions is as a relation between two sets $A$ and $B$, in which every $a \in A$ is related to exactly one $b \in B$ (but not necessarily a distinct one from any other $a' \in A$). To write it in formal symbols, $f: A \to B$ is logically equivalent to $$\Bigl(f \subseteq A \times B \Bigr) \mathbin\& \Bigl( \forall a \in A \,\exists b \in B : \bigl[ (a,b) \in f \bigr] \mathbin\& \bigl[\forall b' \in B: (a,b') \in f \implies b' = b \bigr] \Bigr) .$$ In particular, a function $f: \{ 1, 2 \} \to X \cup Y $ of the type described in your question is a two-element set $\{(1,x), (2,y)\}$ such that $x \in X$ and $y \in Y$. This is different from an element $(x,y) \in X \times Y$.
Why it doesn't matter that they're different
The reason why you should ignore the difference is because these are just two ways that you can reproduce the structure of an ordered pair in set theory. What matters is not how the ordered pairs (or any other particular structure) are built, but rather what you can do with them. For instance, the approach of category theory is to place the emphasis on the mappings of the projections $\pi_1$ and $\pi_2$ which give you the first and second entries of the tuples; and this reproduces everything you really care about for tuples, without fretting about "which set" the tuple is represented by.
Why I have a preference despite the fact that it doesn't matter that they're different
Nevertheless — I must confess a preference (purely aesthetic, mind you) for the definition in terms of functions, if you do choose to spend time contemplating definitions in terms of sets.
If you take the constructions very literally, $A \times B \times C \times D$ has to be interpreted as something like $A \times (B \times (C \times D))$, consisting of tuples $x = (a,(b,(c,d)))$, which is slightly ridiculous. I prefer to think of tuples as $x = (a,b,c,d) := \{(1,a), (2,b), (3,c), (4,d)\}$, which is usually what I really mean (because then $x_1 = a$, $x_2 = b$, etc. is no more than function evaluation).
Furthermore, an infinite product $A \times (B \times (C \times (\cdots))))$ would give rise to sets with infinitely decreasing chains of elementhood relations, which are ruled out by the usual axioms of set theory (specifically the Axiom of Foundation for ZFC or NBG). If you use functions $f: A \to \bigcup_{\alpha \in A} X_\alpha$ to define tuples, it really doesn't matter if $A$ happens to be an infinite set; the set of the tuples is still well-defined (although anyone who disbelieves in the Axiom of Choice may think that it could be empty even if none of the $X_\alpha$ are).
If you spend much time thinking about tuples in terms of sets, defining tuples as functions — and defining functions in terms of "arcs", a word which I choose almost-arbitrarily for the usual set $\{\{a\},\{a,b\}\}$ used to define ordered pairs — makes things a lot nicer.