$\require{begingroup} \begingroup$
$\def\a{\alpha}\def\la{\ln{\a}}\def\e{\mathrm{e}}\def\W{\operatorname{W}}$ $\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$
For $x\in(-\tfrac1{\mathrm{e}},0)$, $\a\in(0,1)$ two real branches of the Lambert W function can be parameterized as follows:
\begin{align} \Wp(x)&= \frac{\a\ln\a}{1-\a} \tag{1}\label{1} ,\\ \Wm(x)&= \frac{\ln\a}{1-\a} \tag{2}\label{2} ,\\ x&=\Wp(x)\exp(\Wp(x)) =\Wm(x)\exp(\Wm(x)) \\ &= \a^{\tfrac a{1-\a}} \ln\Big(\a^{\tfrac a{1-\a}}\Big) = \a^{\tfrac1{1-\a}} \ln\Big(\a^{\tfrac1{1-\a}}\Big) . \end{align}
This parameterization is also consistent with the values of $\Wp,\Wm$ at $x=0$ and $x=-\tfrac1{\mathrm{e}}$: \begin{align} \Wp(0)&= \lim_{\a\to0}\frac{\a\ln\a}{1-\a}=0 ,\\ \Wp(-\tfrac1{\mathrm{e}})&= \lim_{\a\to1}\frac{\a\ln\a}{1-\a}=-1 ,\\ \Wm(0)&= \lim_{\a\to0}\frac{\ln\a}{1-\a}=-\infty ,\\ \Wm(-\tfrac1{\mathrm{e}})&= \lim_{\a\to1}\frac{\ln\a}{1-\a}=-1 . \end{align}
For any $x\in(-\tfrac1{\mathrm{e}},0)$ the value of corresponding parameter $\a$ is found as
\begin{align} \a(x)&=\frac{\Wp(x)}{\Wm(x)} . \end{align}
A dual form of \eqref{1}-\eqref{2} can be used for $\a>1$. In this case
\begin{align} \Wp(x)&= \frac{\ln\a}{1-\a} \tag{3}\label{3} ,\\ \Wm(x)&= \frac{\a\ln\a}{1-\a} \tag{4}\label{4} ,\\ \a(x)&=\frac{\Wm(x)}{\Wp(x)} . \end{align}
One related reference is
D. J. Jeffrey and J. E. Jankowski. "Branch Differences and Lambert W". In: 2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. Sept. 2014, pp. 61-65. doi:10.1109/SYNASC.2014.16
where it is shown that the difference of branches of the Lambert $\W$ function is nontrivial.
Question: is there any other known references, where such parameterization is suggested/used?
Edit Also, in the fundamental paper
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J.Jeffrey, and D. E. Knuth. On the Lambert W Function. Advances in Computational Mathematics, 5:4 (1996) 329-359
there is a closely related example "Solution of a jet fuel problem", Eqns. 2.9-2.10, from which such parameterization can be easily deduced, but still, the parametric form of $\Wp,\Wm$ is not explicitly mentioned.
$\endgroup$
