  We present an improved data-reduction technique to obtain high-precision
proper motions (PMs) of globular clusters using Hubble Space Telescope data.
The new reduction is superior to the one presented in the first paper of this
series for the faintest sources in very crowded fields. We choose the globular
cluster NGC 362 as a benchmark to test our new procedures. We measure PMs of
117 450 sources in the field, showing that we are able to obtain a PM precision
better than 10 $\mu$as yr$^{-1}$ for bright stars. We make use of this new PM
catalog of NGC 362 to study the cluster's internal kinematics. We investigate
the velocity-dispersion profiles of the multiple stellar populations hosted by
NGC 362 and find new pieces of information on the kinematics of first- and
second-generation stars. We analyze the level of energy equipartition of the
cluster and find direct evidence for its post-core-collapsed state from
kinematic arguments alone. We refine the dynamical mass of the blue stragglers
and study possible kinematic differences between blue stragglers formed by
collisions and mass transfer. We also measure no significant cluster rotation
in the plane of the sky. Finally, we measure the absolute PM of NGC 362 and of
the background stars belonging to the Small Magellanic Cloud, finding a good
agreement with previous estimates in the literature. We make the PM catalog
publicly available.
