  The minimal gauged $U(1)_{L_\mu-L_\tau}$ model has long been known to be able
to explain the tension between the theoretical and experimental values of the
muon magnetic moment. It has been explored and tested extensively, pushing the
viable parameter space into a very tight corner. Further, embedding the
$U(1)_{L_\mu-L_\tau}$ model in a supersymmetric (SUSY) framework has been shown
to relax some of these constraints and has recently been shown to explain the
electron anomalous magnetic moment as well. In this model, the logarithm of the
mass ratio of third to second generation (s)leptons control the non-negligible
kinetic mixing and may crucially alter many of the constraints. We confront
both the non-SUSY and SUSY versions of this class of models with the CsI(2017),
the recently released CENNS10 data from the liquid Argon detector as well as
the updated CsI(2020) data of the COHERENT experiment. We use the recoil energy
and timing binned data from CsI(2017) and the energy, time, and Pulse Shape
Discriminator binned data from CENNS10 to find estimates for the model
parameters in a likelihood maximization test. We also show updated exclusions
using all of the above data from the COHERENT Collaboration, as well as
projected exclusions from the ongoing Coherent CAPTAIN-Mills experiment. The
$(g-2)_\mu$ favored values of the $U(1)_{L_\mu-L_\tau}$ gauge coupling that are
still unconstrained overlap with the estimates from COHERENT data within
$1\sigma$. The combined COHERENT data is found to prefer the presence of the
$U(1)_{L_\mu-L_\tau}$ gauge boson over the Standard Model at $\sim1.4\sigma$.
The global minima of a chi-square deviation function using CsI(2020) as well as
CENNS10 total counts has significant overlap with the $(g-2)_{\mu}$ favored
parameter space in the context of the SUSY and non-SUSY $L_{\mu}-L_{\tau}$
models with a mediator mass in the $20-100$ MeV range.
