True weight is gravity's pull, mg; nothing the lift does changes it. A scale never measures gravity: it reads the normal force N it pushes up with, your apparent weight, what your legs feel.
Up is +. ΣF = N − mg = ma, so N = m(g + a): only a counts, never v. Speeding up going up, or slowing down going down: a > 0, N > mg, you feel heavier. Speeding up going down, or slowing down going up: a < 0, N < mg, lighter. Steady speed, up or down: N = mg, so a smooth ride feels like standing still.
At a = −g, N = 0: weightless. Gravity still acts; the floor just falls as fast as you do. Cut the cable and that is the fall, until the emergency brake grips the rails at +g: N = 2mg. You can set a from −g to +g: a floor can only push, and each trip slows as hard as it sped up.
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