Uncertainty about the concept of "product" in category theory This question is motivated by Categories: if $A \times B$ exists then $B \times A$ exists.
To make the question as concrete as possible, let us consider the category of sets. In this category the categorical product is the usual Cartesian product.
Given two sets $X_1, X_2$, one can define the concept of an ordered pair $(x_1,x_2)$ with $x_i \in X_i$. There are various concrete constructions, e.g. $(x_1,x_2) =\{ \{x_1\},\{x_1,x_2\}\}$, but that is irrelevant here. Then $X_1 \times X_2$ is defined as the set of all ordered pairs $(x_1,x_2)$  with $x_i \in X_i$. This definition can be generalized to finitely many $X_1,\ldots, X_n$ and gives the Cartesian product $P = X_1 \times \ldots \times X_n$. We get projections $p_i : P \to X_i$ and can prove that the usual universal property for the categorical product is satisfied.
For an arbitrary family of sets $X_\alpha$ indexed by the elements $\alpha$ of a set $A$, a product of the $X_\alpha$ is usually defined as
$$\prod_{\alpha \in A} X_\alpha = \{f : A \to \bigcup_{\alpha \in A} X_\alpha \mid \forall \alpha \in A : f(\alpha) \in X_\alpha \}. $$
I frequently read that
$$ X_1 \times \ldots \times X_n = \prod_{\alpha \in \{1,\ldots,n\}} X_\alpha .$$
Okay, the definitions of LHS and RHS are somewhat different, so we should more precisely say that there is natural bijection between the LHS and the RHS.
In my opinion the product $X_1 \times \ldots \times X_n$ is defined for a finite ordered index set ($\{1,\ldots,n\}$ with the usual ordering), thus the factors in $X_1 \times \ldots \times X_n$ have an ordering. This is reflected by the sequential ordering of its elements $(x_1,\ldots,x_n)$. In contrast the product $P = \prod_{\alpha \in A} X_\alpha$ is defined for sets $A$ without an ordering. Thus there is nothing like a first or a last factor in $P$.
Here is my question:

Do we have in fact two distinct concepts of categorical products, one for finite ordered index sets and one for arbitrary index sets? Or is this just a nitpick?

Update:
As an example let us consider the set $A = \{ \&,\%\}$. This has no canonical ordering; it allows two orderings which gives two "ordered" products $P' = X_\& \times X_\%$ and $P'' = X_\% \times X_\&$. We can distinguish the first and the second factor, and these are not the same in $P'$ and $P''$. This is an additional information if we compare it with $P = \prod_{\alpha \in A} X_\alpha$. In $P$ there is nothing like a first and and a second factor. Is that difference between $P, P', P''$ relevant in any respect?
 A: In category theory, products are only defined by the universal property. In concrete terms, consider any object $P$ together with morphisms $p_i : P \to X_i$ for $i = 1,..., n$. Then $P$ together with the $p_i$ satisfy the universal property of $X_1 \times ... \times X_n$ iff $P$ together with the $p_i$ satisfy the universal property of $\prod\limits_{i \in \{1, ..., n\}} X_i$. So these two concepts are, in fact, exactly the same concept.
In a given category, we can often pick a "canonical" choice of finite products and a "canonical" choice of arbitrary products. A canonical choice of $n$-ary products for all $n$-tuples of objects in a category will always be unique up to unique natural isomorphism. Similarly, a choice of $I$-indexed products will always be unique up to unique natural isomorphism.
In particular, in the category of sets, the canonical choice of $X_1 \times ... \times X_n$ and the canonical choice of $\prod\limits_{i \in \{1, ..., n\}} X_i$ are, in general, different sets. But there is a natural isomorphism between them, and each one satisfies the universal property of the other.
