A binary operation is a function $f : S\times S\to S$. However, if $0 < \left|S\right| = n < \infty$, we can count the number of such functions:
$$
\#\{f: S\times S\to S\} = \left|S\right|^{\left|S\times S\right|} = n^{n^2}< \infty.
$$
If $S$ is infinite, then there are clearly an infinite number of such functions. So your second conjecture holds if $S$ is infinite, but not if $S$ is finite (if $S = \emptyset$, there is a unique function $f : \emptyset\times\emptyset\to\emptyset$). Your first conjecture is true: if $S\neq\emptyset$, then say $a\in S$. Define $f : S\times S\to S$ to be the constant function sending any element of $S\times S$ to $a$. This is a binary operation for any nonempty set, and we have already seen a binary operation for $\emptyset$ (your argument for nonempty $S$ also works).
The cartesian product of two sets : $X,Y$ is a set $Z$ defined as :
$Z = \{ (x,y) \, | \, x \in X \, \text {and} \, y \in Y \}$
where $(x,y)$ is the ordered pair having $x$ as first component and $y$ as second component.
Thus, the cartesian product $X \times Y$ is the set of all ordered pairs with first component in $X$ and second component in $Y$.
A relation $R$ with domain in $X$ and range in $Y$ is a subset of the cartesian product $X \times Y$, i.e. :
$R \subseteq X \times Y$.
Thus, a relation is a set of ordered pairs.
A function $F$ is a relation satisfying the ("functionality") condition :
if $(x_1,y_1) \in F$ and $(x_1,y_2) \in F$, then $y_1=y_2$.
A binary operation $f : Y \times Y \to Y$ is a function from the cartesian product $Y \times Y$ to the set Y, i.e. a subset of $(Y \times Y) \times Y$, because it "maps" an ordered pair $(y_1,y_2)$ into an element $y_3$, with $y_i \in Y$.
You can try to clarify the definitions with some simple examples.
Let $\mathbb N = \{ 0, 1, 2, ... \}$ the set of natural numbers.
Consider the cartesian product $\mathbb N \times \mathbb N$ and :
the relation $<$ ("Less then"), i.e. $(n,m) \in L$ iff $n < m$,
the function $s$ ("Successor"), i.e. $(n,m) \in S$ iff $m=s(n)$
the (binary) operation $+$ ("Plus"), i.e. $((x,y),z) \in P$ iff $z=x+y$.
Best Answer
A binary operation on $S$ is a function $S\times S\to S$. You are correct.