  Recently, several eclipsing millisecond pulsars have been shown to experience
strong and apparent weak lensing from the outflow of their ionized companions.
Lensing can be a powerful probe of the ionized plasma, with the strongest
lenses potentially resolving emission regions of pulsars. Understanding lensing
in the `laboratory-like' conditions of an eclipsing pulsar may be analogously
applied to fast radio bursts, many of which reside in dense, magnetized
environments. We examined variable dispersion measure (DM), absorption,
scattering, and flux density in the original Black Widow pulsar PSR B1957+20
through an eclipse at the Arecibo Observatory at 327 MHz. We discovered clear
evidence of the two regimes of lensing, strong and apparent weak. We show that
the flux density variations in the apparently weak lensing regime can be
modeled directly from variations of DM, using geometric optics. The mean
effective velocities in the ingress, $954\pm 99$ km/s, and egress $604\pm 47$
km/s cannot be explained by orbital motions alone, but are consistent with
significant outflow velocity of material from the companion. We also show that
geometric optics can predict when and where the lensing regime-change between
weak and strong occurs, and argue that the apparent weak lensing is due to
averaging many images. Our framework can be applied in any source with variable
electron columns, measuring their relative velocities and distances. In other
eclipsing pulsars, this provides a unique opportunity to measure companion
outflow velocity, predict regions of weak and strong lensing, and in principle
independently constrain orbital inclinations.
