There are three different cases where e=1. One is a parabola (zero total energy, non-zero angular momentum). Another is a degenerate parabola (zero total energy, zero angular momentum). The third is a degenerate ellipse (negative total energy, zero angular momentum). Some jokingly call this case an "orthogonal orbit". The rock falling from at rest represents an example of this third case.
Think of it in terms of a limit. Suppose that instead of starting with zero velocity at the Moon's orbital distance, the rock starts with zero radial velocity and a tiny but non-zero tangential velocity. The rock will follow an elliptical orbit with apogee at the Moon's orbital distance. In the limit that that tangential velocity goes to zero, the eccentricity will go to one. As Kepler's equations become singular at this limit, you cannot use Kepler's equations directly. However, you can use Kepler's equations to solve the problem for any non-zero angular momentum. Take the limit of this solution as angular momentum goes to zero.
These machinations would have come in handy had you been asked to determine the time needed to fall from 384,400 km to 96,000 km. However, you were not asked that. You were just asked to determine the velocity. All that takes is conservation of energy considerations. Use the approach to which you alluded in post #3.