In the vomit comet lift, the reaction of the lift on the man is zero and the weight is mg. Theyre two completely different forces. The reaction forms a force pair at your feet and the weight is applied through your COM so effectively your stomach. Drawing a diagram of the person, the lift applies R upwards at your feet and gravity applies a force downwards (weight) at your COM. You resolve vertically and difference the two (opposite directions) to get the net force.
If youre in a stationary lift (or moving with constant velocity), gravity is exerting a force mg on the person (their weight). There is zero acceleration so newton 2 on the person gives
m*0 = weight - reaction = mg - reaction
reaction = mg = weight. Which is what most people think of as the "usual" reaction.
However, if both the lift and the person was accelerating downwards at g/2, then youd have for the person
ma = weight - reaction
reaction = mg - mg/2 = mg/2
so the reaction is half the previous value.
Alternatively if you were accelerating upwards at g/2 so a=-g/2 (positive downwards), then youd have (for the person) similar to above
reaction = mg + mg/2 = 3mg/2
so the floor is pushing you up harder than gravity is pulling you down and the net force and acceleration is upwards. If youre in a rapidly accelerating lift (up or down) you physically feel this effect.