  We perform a detailed study of neutrino-electron elastic scattering using the
mono-energetic $^{7}$Be neutrinos in Borexino, with an emphasis on exploring
the differences between the contributions of $\nu_e$, $\nu_\mu$, and
$\nu_\tau$. We find that current data are capable of measuring these components
such that the contributions from $\nu_\mu$ and $\nu_\tau$ cannot be zero,
although distinguishing between them is challenging -- the differences stemming
from Standard Model radiative corrections are insufficient without
significantly more precise measurements. In studying these components, we
compare predicted neutrino-electron scattering event rates within the Standard
Model (accounting for neutrino oscillations), as well as going beyond the
Standard Model in two ways. We allow for non-unitary evolution to modify
neutrino oscillations, and find that with a larger exposure (${\sim}30$x),
Borexino may provide relevant information for constraining non-unitarity, and
that JUNO may be able to accomplish this with its data collection of $^{7}$Be
neutrinos. We also consider novel $\nu_\mu$- and $\nu_\tau$-electron scattering
from a gauged $U(1)_{L_\mu - L_\tau}$ model, showing consistency with previous
analyses of Borexino and this scenario, but also demonstrating the impact of
uncertainties on Standard Model mixing parameters on these results.
