Terminology of "commuting" diagrams. The question is just about terminology. "Commuting diagrams" generalize both associativity and commutativity laws. They don't seem specifically related to commutativity, so why do we say the diagram "commutes"?
Clarification:
Both "commutativity" and "associativity" are interchange laws. The first says path ab can be interchanged with path ba. The second says that the path (gf)h and the path g(fh) can be interchanged. It seems arbitrary that the name of only one of these interchange laws (i.e. "commutativity") was chosen to describe the general case.
 A: Actually, it is quite related to commutativity.
$\require{AMScd}$
\begin{CD}
A@>{f}>> A\\
@V{g}VV @VV{g}V\\
A @>{f}>> A
\end{CD}
Let $f,g$ be endomorphisms of some algebraic structure $A$. Then $f\circ g=g\circ f$ if and only if the diagram above commutes. So, in some cases commutativity of these operators is equivalent to commutativity of the diagram.
Anyway, this is my interpretation.
Edit: To address the comment, I will only mention that associativity is also a sort of commutativity. For instance, we could define associativity to be the commutativity of the following diagram:
\begin{CD}
A\times A\times A@>{\mu\times \operatorname{Id}}>> A\times A\\
@V{\operatorname{Id}\times \mu}VV @VV{\mu}V\\
A\times A @>{\mu}>> A
\end{CD}
where $\mu:A\times A\to A$ is given by $\mu(x,y)=x\cdot y$, and $\cdot$ is a binary operation on $A$. Then the commutativity of the diagram says that for all $x,y,z\in A$
$$x\cdot(y\cdot z)=(\mu\circ (\operatorname{Id}\times \mu))(x,y,z)=(\mu\circ (\mu\times \operatorname{Id}))(x,y,z)=(x\cdot y)\cdot z.$$
So, it is reasonably to consider associativity as a "special case" of a commutativity condition.
A: The etymology of the word "commute" comes from "to interchange two things", from Latin. When we talk about, say, a group commuting, we talk about interchanging the orders of the elements ($ab = ba$). When we talk about a graph commuting, we talk about interchanging the paths we travel through the graph ($f = g \circ h$).
A: If $\mathcal{M} = (M, \cdot)$ is a monoid, regarded as a category with a single object $\star$ (so the morphisms of the category are the elements of the monoid and the composition is given by the monoid operation), then two elements $a,b \in M$ commute (in the sense that $a \cdot b = b \cdot a$) if and only if the following diagram commutes:
$$
\begin{array}{ccc}
\star & \xrightarrow{a} & \star \\
{\scriptsize b} \downarrow ~ && ~ \downarrow {\scriptsize b} \\
\star & \xrightarrow[a]{} & \star
\end{array}
$$
So in this sense, commutativity for diagrams generalises the usual notion of commutativity.
