I did not know where to put this answer, whether here or in this thread. Please, consider it as a remark on a possible definition of $L_\infty$-homotopies. I would like to summarize some considerations on $L_\infty$-algebra homotopies using as prototype the homotopies of $A_\infty$ morphisms.
With $A$ and $B$ we denote $\mathbb Z$-graded vector spaces on a ground field of characteristic zero (for simplicity). In what follows, a graded vector space will be also called "graded object".
- A little remark: the chain complexes case
If $A$ resp. $B$ are chain complexes, with differentials $d_A$, resp. $d_B$, and $f,g: A\rightarrow B$ are (chain complexes) morphisms then a homotopy $H: A\rightarrow B$ between them is a degree $-1$ morphism of graded objects s.t. $f-g = d_BH+Hd_A$. No extra condition on $H$ is needed, as $d_B (f-g) = (f-g) d_A$ is automatically satisfied.
- $L_\infty$ case
Let $(A,D_A)$ resp. $(B,D_B)$ by $L_\infty$-algebras; with $D_A$, resp. $D_B$ we denote the squaring to zero, $+1$ degree coderivations on the free graded co-commutative coalgebras over $A$, resp. $B$. These latter are denoted by $(C(A),\Delta_A)$ resp. $(C(B),\Delta_B)$: the $\Delta$'s are the coproducts.
Let $F,G$ be $L_\infty$-morphisms from $A$ to $B$, i.e. differential graded coalgebra maps $F,G:(C(A),D_A,\Delta_A)\rightarrow (C(B),D_B,\Delta_B)$ satisfying the known compatibility conditions w.r.t. coderivations and coproducts.
Definition I An $(F,G)$-derivation $R$ of degree $k$ is a graded object map $R:C(A)\rightarrow C(B)$ of degree $k$ s.t.
$$\Delta_B H = (F\otimes H+ H\otimes G)\Delta_A.~~ (**) $$
We arrive at the main definition of this answer, i.e.
Definition II A homotopy $H$ between the $L_\infty$-algebras morphisms $F$ and $G$ is an $(F,G)$-derivation of degree $-1$ s.t. $F-G = D_BH+HD_A$.
The questions are: why do we impose the $(F,G)$-coderivation relation (**) ? Which consequences does it have? The (inevitably partial) answers to these questions are my motivation for such definition of $L_\infty$-homotopies. In summary:
As $H$ is an $(F,G)$-coderivation, then it satisfies a lifting property and it is uniquely determined by its components $H_n: A^{\otimes n}\rightarrow B$. $H$ is lifted with the well-known formulae $H=\sum F_{\bullet}^{\otimes\bullet}\otimes H_{\bullet}\otimes G_{\bullet}^{\otimes\bullet}$.
$D_BH+HD_A$ is an $(F,G)$-coderivation ($D_A$ and $D_B$ are $(1,1)$-coderivations) as well; this implies that the relation $F-G = D_BH+HD_A$ can be expressed with the familiar tower of $L_\infty$-relations found in the literature.
$G + D_BH+HD_A$ and $F-D_BH+HD_A$ really cocommute with coproducts, as $L_\infty$-algebra morphisms should.
In summary, I believe a definition of homotopy of $L_\infty$-algebras could be the one given in definition II, above. As said in the introduction, similar constructions take place in the realm of $A_\infty$-algebras: in this setting one uses as graded coalgebras the reduced tensor coalgebras. A good reference can be the first chapter in Hasegawa's PhD Thesis.
