Is the lift vector reversed during inverted flight?
This is easiest to understand with a fully symmetrical wing, which indeed are popular with aerobatic aircraft.  The direction of "lift" has nothing to do with sky or ground and everything to do with your angle of attack to the relative wind.

When inverted, you simply (as the professor said) push forward on the stick, which makes the wind strike the "top" of your wing.  Inverted, this does the trick as lift is now towards your floor, but gravity is toward your head.

Right side up, you make the wind strike the bottom of your wing by pulling back on the stick.