  We present a detailed modeling and computation methodology to determine the
optical Cherenkov signals produced by upward-moving extensive air showers
(EASs) induced by $\tau$-leptons and muons, sourced from the interaction of
high-energy astrophysical neutrinos interacting in the Earth. Following and
extending the physics modeling and Cherenkov signal simulations performed in
arxiv:1902.11287, this scheme encompasses a new, state-of-the-art computation
of the muon neutrino propagation inside the Earth and the contribution to the
$\tau$-lepton muon decay channel. The modeling takes into account all possible
$\tau$-lepton decay and muon energy loss channels that feed the optical
Cherenkov emission, produced by both tau and muon initiated EASs. The EAS
modeling uses the electron energy, angular, and lateral distributions in the
EAS and their evolution as well as the wavelength dependence of the Cherenkov
emission and its atmospheric attenuation. The results presented here are
focused on the detection capabilities of sub-orbital (balloon-borne) and
orbital (satellite) based instruments. The latter case was calculated for
POEMMA to compare to that presented in arxiv:1902.11287, specifically including
the muon-decay channel of $\tau$-leptons and the muonic EAS Cherenkov signal
from muon neutrino interactions in the Earth. By detailing all these individual
contributions to the optical Cherenkov emission and detection, we show how the
ensemble that includes muonic channels provides a large detection capability
for space-based, high-energy cosmic neutrino detection. Specifically, we show
that for neutrino energies $\lt$ 10 PeV, the upward-EAS sensitivity due to muon
neutrino interactions in the Earth begin to dominate over that for tau neutrino
interactions, effectively extending the neutrino sensitivity to lower energies.
