Differences between groups and rings when understood as categories A group can be understood as a category with a single object $*$ whose hom-set $\text{hom}(*,*)$ forms a group with composition as the group multiplication.
A ring can be understood as a category that is preadditive with a single object $*$. Preadditive means that for a category $C$, $\text{hom}_C(A,B)$ forms an abelian group that is distributive with respect to composition of arrows.
The group structure appears in different ways in these two categories. It seems weird to me that the group multiplication is the composition of arrows for groups as categories but for rings it is some binary map of arrows in the same hom-set. Why is this the 'best' way to understand these structures as categories? Is there some way to understand a ring as a category where composition of arrows forms a group (like the categorical sense of a group)?
 A: I think the basic way to understand a ring as a category is as a monoid in the category of abelian groups.
On one hand this says it is one abelian group $A$ and abelian group homomorphisms $Hom(A,A)$. But if you look at what it says when you identify the elements of $Hom(A,A)$ as "elements of the ring," it just says everything about distributivity of multiplication over addition.  The category axioms provide for the associativity of multiplication in the ring monoid.
A group is a monoid in the category of sets with the extra property that its arrows are isomorphisms.
So both are realized as monoid over different categories, and the group specializes by having extra requirements on its arrows.  The closest analogue for a ring is a division ring, of course, where all its nonzero arrows are isomorphisms.
Added
Here are the details for what a monoid in a category is.. It is more involved than I remembered but it is worth working through as it provides a convenient viewpoint for many objects including rings, algebras, graded algebras, topological monoids etc.
A: Instead of comparing groups and rings, one needs to compare monoids and rings:

*

*An one-object category is the same as a monoid.

*An one-object preadditive category is the same as a ring.

We see from this that the multiplication in a monoid needs to be compared to the multiplication in a ring.
The addition in rings has no corresponding structure in monoids.
Every ring has an underlying additive group, and some monoids happen to be groups. But that doesn’t matter for the above observation:
the multiplication in a group is a special case of the multiplication in a monoid, and therefore needs to be compared to the multiplication in a ring (and not the addition).
The addition in rings has no corresponding structure in groups.
A: I don’t know if this quite answers your question, but the point you focus on as your dissatisfaction seems to me to be a misunderstanding:

It seems weird to me that the group multiplication is the composition of arrows for groups as categories but for rings it is some binary map of arrows in the same hom-set.

Viewing a ring as a one-object pre-additive (i.e. $\newcommand{\AbGp}{\mathrm{AbGp}}\AbGp$-enriched) category, its multiplication is precisely composition of arrows, exactly like in groups-viewed-as-one-object categories.  The point is that in single-object categories (plain or enriched), composition is exactly “some binary map of arrows in the same hom-set”, $\newcommand{\C}{\mathcal{C}}\C(*,*) \otimes \C(*,*) \to \C(*,*)$.
So I’d say the picture you set out is indeed a very good way to look at them:

*

*monoids (so in particular, groups) are 1-object categories;

*rings are 1-object categories enriched in $(\AbGp,\otimes)$ (aka monoids in $(\AbGp,\otimes)$);

*the forgetful functor $(\AbGp,\otimes) \to (\mathrm{Set},\times)$ is lax monoidal; so it sends monoids in $(\AbGp,\otimes)$ to monoids in $(\mathrm{Set},\times)$, sending a ring to its underlying multiplicative monoid.

