How to define the equivalence of Maurer-Cartan elements in an $L_{\infty}$-algebra?

To add a bit to what Damien says, addressing your question on how to generalise the gauge approach (which is equivalent to the approach outlined by Damien, as proved by several people):

You can view gauge symmetries in DGLAs via solving the differential equation $$ \frac{d\alpha}{dt}=-\partial\xi-[\alpha,\xi], $$ where $\xi$ is the given element of degree $0$. This generalises to homotopy Lie algebras as follows: consider the differential equation $$ \frac{d\alpha}{dt}=-\partial\xi-[\alpha,\xi]-\frac12[\alpha,\alpha,\xi]-\ldots-\frac{1}{p!}[\underbrace{\alpha,\ldots,\alpha,}_{p \text{ times}}\xi]_{p+1}-\ldots, $$ where the right hand side is simply the negative of $[\xi]_1^\alpha$, the first structure map of the twisted Lie-infinity structure $$ [x_1,\ldots,x_k]_k^\alpha:=\sum_{p\ge0}\frac{1}{p!}[\underbrace{\alpha,\ldots,\alpha,}_{p \text{ times}}x_1,\ldots,x_k]_{k+p}. $$ From that it is almost obvious that moving along the integral curves of this equation preserves the property of being Maurer--Cartan, since the Maurer--Cartan condition for $\alpha+\beta$, where $\alpha$ is a Maurer--Cartan element, and $\beta$ is infinitesimal becomes $$ \partial\beta+[\alpha,\beta]+\frac12[\alpha,\alpha,\beta]+\ldots+\frac{1}{p!}[\underbrace{\alpha,\ldots,\alpha,}_{p \text{ times}}\beta]_{p+1}+\ldots, $$ that is $[\beta]_1^\alpha=0$, and so $\beta=[\xi]_1^\alpha$ satisfies that, $[\cdot]_1^\alpha$ being a differential of the twisted structure. This circle of ideas is explained in many places, one important reference is ``Lie theory for nilpotent $L\_\infty$-algebras'' by Ezra Getzler (Ann. of Math. (2) 170 (2009), no. 1, 271--301.).

This is explained in Section 4.5.2 of "deformation quantization of poisson manifolds" by Kontsevich (http://arxiv.org/abs/q-alg/9709040).

The way you wrote the homotopy between two Maurer-Cartan elements is not enough : as it is explained in the above reference you also need a 1-parameter family of infinitesimal gauge equivalences.

A quick reformulation of Kontsevich definition is the following. An equivalence between two Maurer-Cartan elements $a$ and $b$ in $\mathfrak g$ is a Maurer-Cartan element $c$ in $DR([0,1])\otimes\mathfrak g$ such that $a=c(0)$ and $b=c(0)$.

Note that $DR(...)$ stands for the de Rham algebra of "...".