When we are discussing a binary operation $*:X \times X \to X$, we typically say that $*$ is commutative if $*(x,y) = *(y,x)$ for all $x,y \in X.$ However, when discussing a function $F: X \times X \to Y$ (where $X \neq Y$) such that $F(x,y) = F(y,x)$ for all $x,y \in X,$ I often hear $F$ called "symmetric" - as in the case of a metric or an inner product.
What exactly is the difference between the words "commutative" and "symmetric" in describing a function? Is the above the full story?
I wouldn't use the term commutative to describe a binary operation unless it were also associative. In that case, commutativity can be rephrased in a particular way that doesn't at all apply to functions $X \times X \to Y$. But first I need to describe how associativity can be rephrased in this way.
First, there's no reason to privilege binary operations. In practice we don't only, say, multiply two numbers; we often multiply three or more numbers. To say that a binary operation $a \ast b$ is associative is precisely to say that it comes from a compatible family of $n$-ary operations $a_1 \ast a_2 \ast ... \ast a_n$ in a particular way; compatible means, in particular, that
$$(a_1 \ast ... \ast a_n) \ast (a_{n+1} \ast ... \ast a_m) = (a_1 \ast ... \ast a_m).$$
Said another way, associativity really means that I have a nice way to eat a list of elements and spit out an element in a way which is compatible with concatenation of lists.
An operation with the above property which is in addition commutative can be thought of as eating, not a list, but a multiset of elements and spitting out another element; that is, order doesn't matter. Being able to interpret commutativity in this way requires that I be able to write down $n$-fold compositions so it doesn't apply to functions $X \times X \to Y$.
The only difference I can see between the two terms is that commutativity is a property of internal products $X\times X\to X$ while symmetry is a property of general maps $X\times X\to Y$ in which $Y$ might differ from $X$.
I'd say that describes the difference in most common usage, and that there is really nothing more to it. There are exceptions, due to history. For example, if $A$ is an algebra, then we sometimes sat that a bilinear map $f:A\times A\to k$ to the base field is associative if $f(xy,z)=f(x,yz)$, and $f$ is not an internal product...
A set is a collection of elements. A 'relation is a set' of ordered tuples. A 'function is a relation' f(x)=y with each x pointing to a unique y.
A 'binary relation' is a set of ordered 'pairs'. An n-ary operation is an example of function. A 'binary operation' is a 2-ary operation.
symmetric is the property of the binary relation. xRy = yRx. commutative is the property of the binary operation. f(x, y) = f(y, x)
That is about sets, relations, and functions. Now let us talk about the dot product (a case of inner product).
The n-ary/binary operation has the closure property. So, it has to be f(X,X) -> X. It cannot be f(X,X) -> Y. So, Dot Product is not a binary operation since the product of two vectors is a scalar. Vector product, though, is a binary operation.
The use of the word 'symmetric' for an inner product relates to the interchangeability of the vector arguments. It feels like 'commutative', but then we are not in the realm of a binary operation.
