Why are algebraic structures typically defined with operators/relations and laws? Here is a definition for equivalence relations:

*

*An equivalence relation on a set $X$ is a relation $\sim$ which is reflexive, symmetric, and transitive.

As compared to:


*An equivalence partitioning is a function $[\cdot] : X \to 2^X$ from elements of $X$ to subsets of $X$. For every $x$, we refer to $[x]$ as the class of $x$. We also require that (A) elements are in their own classes, i.e., $x \in [x]$, and (B) classes are disjoint, i.e., $y \in [x] \implies [y] = [x]$.
If two elements $x, y$ are in the same class, we write $x \sim y$. (Reflexivity, symmetry, and transitivity follow as trivial lemmas from this definition, perhaps even not worth the time to note.)


*An equivalence relation on a set $X$ is an undirected graph over $X$ where each component is complete.
For an element $x \in X$, we write $[x]$ to denote the (elements of the) component containing $x$.
If two values $x, y$ are in the same component, we write $x \sim y$.


*A partitioning on a set $X$ is a set $P$ of nonempty subsets of $X$ where (A) elements of $P$ are disjoint; and (B) $\bigcup P = X$.
Note that for an element $x \in X$, there is a unique element of $P$ that contains $x$. We call this the class of $x$ and denote it as $[x]$.
If two elements $x, y \in X$ are in the same class, we write $x \sim y$.
In my experience, definition (1) is by far the most used, despite the others being equivalent (afaik) and massively more mentally evocative. My mental model of equivalence relations is that they are fundamentally about partitions, and definition (1) obscures this fact while the others bring it to light.
My question is: if definitions (2), (3), and (4) are equivalent to (1) but more appealing to the intuition (or, if I dare, to what equivalence relations are "actually" about), then why is (1) always used?
Is it:

*

*That definitions (2), (3), and (4) aren't universally more appealing to the intuition; most people actually prefer (1)?

*Just a historical accident? If graphs had come around before symbols, would definition (3) be preferred?

*Because equivalence relations are often used in foundational mathematics, where other definitions wouldn't work?

*Just that definition (1) is most convenient for proofs? (Is that a historical accident?)

*That mathematicians just don't care this much about an appeal to intuition; once the concept is learned, the definition ceases to matter?

*Because mathematicians care about simple definitions, and definition (1) is somehow simpler than the rest?

*That there are many ways to view equivalence relations, and each definition will appeal to a different viewpoint, and considering one viewpoint "better" than another is mistaken?

For what it's worth, this experience is not remotely limited to equivalence relations for me. I have similar thoughts about preorders, partial orders, total orders, semigroups, monoids, etc.
 A: 
Why are algebraic structures typically defined with operators/relations and laws?

What are algebraic structures if not that?  From the wiki:

In mathematics, an algebraic structure consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A of finite arity (typically binary operations), and a finite set of identities, known as axioms, that these operations must satisfy.

So you are sort of asking "why do we define algebraic structures that way?"  Some may find a deeper answer but I feel like there is not much more to this than "because it is useful to have a term for 'set with operations/identities'"
Based on the examples you gave, you rather intended the question to be something of the sort "why are some definitions in abstract algebra phrased in terms of operations or relations while others aren't?"
My response is, I think, "What do you mean? Everything you wrote used a relation and your example mentions no operations..."
In other words, you haven't demonstrated that there is a salient difference between 1 and 2, 3, 4.  Most people are going to think you are just saying the same thing in four different ways (nothing wrong with that) but that there is no real distinction.  Relations are fundamental tools in mathematics, sometimes so much so that we forget that we talk about relations all the time (e.g. functional relations, aka functions.)
Numbered list, point by point
In 1), you define a very special relation with axioms that focus on elements.  This is very concrete, and emphasizes the special properties of the relation. Maybe this focus on elements is what you're actually interested in, because that does set this apart from the final three.  To most people the utility of reasoning with elements is self-evident, but some mathematicians will also remind us that sometimes focusing on elements is too myopic, and that much can be learned from a perspective that avoids elements altogether.
In 2), you describe an equivalence relation using a function (which is a relation) and hide some of the element-axioms within conditions on sets.
In 3), you describe an equivalent relation as a graph, which is often defined as two sets (nodes and edges) and two functions (src and dst) which tell you where each edge starts and stops (relating nodes to edges.) Some of the axioms about elements are again hidden under the rug of definitions, but still if you unwind them all you get the same thing.
In 4) you perhaps get the furthest away from relations in general.  You define the partition purely in terms of subsets of a set and conditions on their intersection and union.  But still, we haven't escaped relations. "Being an element of" is a primitive relation between sets, so when you say "each element is in a class" you are implicitly using a relation. When you say "two classes are disjoint or equal" you are using another relation (equality)
So I would say you have perhaps not appreciated that relations permeate all of mathematics through set theory and/or category theory, and that there is pretty much no such thing as a "definition that doesn't use relations in some form or another."  At the very least one can say it is pretty much impossible to avoid them in algebra.
Addressing the bulleted list
Most of the bulleted list I do not think is worth retaining, but there are two that ring true:

*

*Just that definition (1) is most convenient for proofs?

*That there are many ways to view equivalence relations, and each definition will appeal to a different viewpoint, and considering one viewpoint "better" than another is mistaken?

I've mentioned a slogan like this in a previous post:

Part of learning mathematics is about layering intuitions

The point is that multiple intuitions provide more insight than just the sum of their individual content.  So for that reason, you and I are in accord on the second bullet point. That is something I can get on board with.
The first bullet point is true enough: sometimes you have definitions that are formulated specifically to make proofs easier. That is part of what a good definition does.
When multiple definitions exist it is usually because they each contain their own unique perspective on the thing being studied.  In such a case it is a mistake to declare that one is somehow categorically "less intuitive" because intuition is largely subjective, and different definitions are going to be useful to different audiences.
