Why do counits go that way? Imagine you want to motivate for an audience the definition of an adjunction in terms of unit and counit. So you can say: Often two functors
$\mathcal{C} \begin{array}{c} \stackrel{\large F}{\rightarrow} \\ \stackrel{\leftarrow}{G}\end{array} \mathcal{D}$ are not really inverse, up to isomorphism, but still, the compositions $F \circ G$ and $G \circ F$ are somehow "connected" to the identity functors. So we have natural transformations
$$\eta : \mathrm{id}_{\mathcal{C}} \to G \circ F,$$
$$\varepsilon : F \circ G \to \mathrm{id}_{\mathcal{D}},$$
which should be compatible in a suitable way. Then someone says:

"Wait, why not taking $\varepsilon : \mathrm{id}_{\mathcal{D}} \to F \circ G$ instead? What's wrong with that?"

What would you answer?

Just to be clear: I understand the concept of an adjunction and know that $\varepsilon$ has to go from $F \circ G$ to $\mathrm{id}_{\mathcal{D}}$, but I am not sure how to explain this  to a novice without going into the details of the notion of an adjunction, for example the triangle identities or the alternative hom-set-definition, which by the way I would like to exclude from the discussion here. A good answer should be intuitive.

Notice that it is not true that we cannot state any compatibility conditions between two natural transformations $\eta : \mathrm{id}_{\mathcal{C}} \to G \circ F$ and $\varepsilon : \mathrm{id}_{\mathcal{D}} \to F \circ G$. An obvious candidate would be
$$F \circ \eta = \varepsilon \circ  F : F \to F \circ G \circ F$$
and/or
$$\eta \circ  G = G \circ  \varepsilon : G \to G \circ F \circ G.$$
By the way, does this concept have a name?
The unit-counit definition of an adjunction makes sense in any 2-category, and so it makes sense in any monoidal category, where it recovers the definition of a dualizable object. So you can motivate this somewhat more general notion using the familiar special case of vector spaces: when a vector space $V$ has a dual $V^{\ast}$, there is a unit map $1 \to V^{\ast} \otimes V$, namely the coevaluation, and also a counit map $V \otimes V^{\ast} \to 1$, namely the evaluation. 

Of course it's hard to distinguish between the two possible orders of the tensor product here since $\otimes$ is symmetric monoidal. At this point I would draw the triangle identities using string diagrams, which justifies using the orderings above, although if you're going to both avoid that and also the homset definition of an adjunction I don't know what to tell you.
Firstly, as tcamp also commented, there also exists a similar construction as adjunctions in any bicategory where the 'unit' and 'counit' go in the same direction, and these are the (dual) Morita contexts, and it is also worth to study, having already wide literature, at least in its origin world of rings. 
(Note that adjoint equivalences are immediately Morita [and dual Morita] contexts by flipping the [co-]unit.)

Secondly, consider the two (pseudo) embeddings of bicategories ${\bf Cat}\to{\bf Prof}$ sending $F\mapsto F_*$ and $F\mapsto F^*$, where $F_*=(a,b)\mapsto\hom_B(Fa,b)$ and $F^*=(b,a)\mapsto\hom_B(b,Fa)$ for a functor $F:A\to B$.
Observe that both of these are contravariant in exactly one dimension ($F\mapsto F^*$ on arrows and $F\mapsto F_*$ on 2-cells).

Moreover, we can roll these two maps in one embedding of double categories ${\bf Q}({\bf Cat})\to{\bf Q}({\bf Prof})$ which is contravariant in exactly one direction, where ${\bf Q}(B)$ means the double category of Ehresmann quintets of the bicategory $B$. 
Consequently, orthogonal adjoint pairs in ${\bf Q}({\bf Cat})\ $ (i.e. adjoint functors with counit-unit-definition) correspond to companion pairs in ${\bf Q}({\bf Prof})\ $ (which yields the "$L_*\simeq R^*$" definition).