I think one can clarify this in a clear-cut way. Partially following OP's terminology, I'll call E-structure a set $X$ endowed with three binary laws $\sigma=\cdot$, $\lambda$, $\rho$ satisfying the given axioms 1,2. I'll write $\sigma(x,y)=xy$. It is a W-structure if moreover it satisfies the 3rd axiom (depending only on $\sigma$), namely $(xy)(zt)=(xz)(yt)$.
Let me now consider this as a magma $(X,\sigma)$ and then discuss existence and uniqueness of $\lambda,\rho$.
A first remark is that in the presence of a unit, Axiom 3 implies commutativity. Also (commutativity + associativity) implies Axiom 3, so Axiom 3 for a monoid means commutative.
The existence of $\lambda$ means that $(xy,y)=(x'y',y')$ implies $x=x'$. That is, $xy=x'y$ implies $x=x'$. This just means that the magma $(X,\cdot)$ is right-cancelative. Similarly the existence of $\rho$ means left-cancelative, and
For a magma $(X,\cdot)$, there exists an E-structure with $\sigma$-law is the magma product $\cdot$ iff $(X,\cdot)$ is cancelative. (And it is an W-structure iff it satisfies identically $(xy)(zt)=(xz)(yt)$.
About uniqueness: clearly $\lambda$ is uniquely determined on the image of the map $X^2\to X^2$, $(x,y)\mapsto (xy,y)$, and can be arbitrarily modified on the complement of this image $I=\bigcup_{y}Xy\times\{y\}$. A similar thing happens for $\rho$, with $J=\bigcup_x\{x\}\times xX$. So an E-structure (resp. W-structure) is a cancelative magma, along with some choice of maps on these complements (which seem not to be the point of interest, as OP focusses on the law $\cdot=\sigma$). (Nevertheless, modifying $\lambda$ and $\rho$ might affect direct indecomposability.)
Actually $I=X^2$ iff $Xy=X$ for all $y$, etc. Hence, the pair $(\lambda,\rho)$ is at most unique if and only if all left and right multiplications in the magma $X$ are surjective. As seen above, its existence means they're injective. So the existence and uniqueness of $(\lambda,\rho)$ means that left and right multiplications are bijective. (For a semigroup, this means being a group.)
OP's theorem, as stated in the post can be restated as: if $(X,\cdot)$ is a cancelative commutative idempotent magma, then it admits a structure of $\mathbf{Z}[1/2]$-module such that $xy=\frac12(x+y)$ for all $x,y$. [This is precisely equivalent to the formulation in the post. OP claims in a comment that the result is weaker, namely that every cancelative commutative idempotent magma embeds into another one that admits such a structure of $\mathbf{Z}[1/2]$-module inducing the magma law as average map.]
Update: OP has added axioms $\lambda(yx)x=y$, $x\rho(xy)=y$. These additional axioms just mean that right and left multiplications are not only injective, but bijective. For finite magmas this doesn't change anything, we get cancelative ones. For infinite ones of course this changes: for instance in the associative case, we get the groups instead of the semigroups. It also discards Jónsson algebras (that is magmas for which the law $X\times X\to X$ is bijective), which satisfied the original axioms 1,2 among E-structures. Of course it also changes the notions of free E-structure and free W-structures. Last and not least, it ensures that $(\lambda,\rho)$ is determined by the law (i.e., by $\sigma$).
Eventually, with these axioms the OP's E-structures are known as quasigroups and the W-structures are known as medial quasigroups (thanks to user zeb for the terminology). The Bruck–Murdoch-Toyoda theorem (same link, also mentioned by user zeb) says that a medial quasigroup is always of the form $(A,\ast)$ defined from an abelian group $(A,+)$, for some commuting pair of automorphisms $(\varphi,\psi)$ of $A$ and constant $c\in A$, with $x\ast y=\varphi(x)+\psi(y)+c$.
(This is quite restrictive: for instance, it implies that for any two $x,x'$, writing $L_x(y)=xy$, we have $L_x^{-1}L_{x'}(y)=y+\psi^{-1}(\varphi(x)-\varphi(x'))$, so that $L_{x}^{-1}L_{x'}$ is a translation. In particular, if $A$ is finite with $|A|\notin 4\mathbf{Z}+2$, all permutations $L_x$ have the same signature.) Similarly for right $\ast$-multiplications.