I know very little about driving technique. But my understanding of what we did that day is as follows:
With lift throttle oversteer, it seems like you would lose speed because now only your momentum is carrying you forward, and you would be experiencing engine braking in addition to friction and be slowing down. By feathering the brake pedal with left foot braking, you acheive the same forward shifting of weight which both lets the rear slide around, but also gives the front more traction. Since you never released the gas, your front wheels are always pulling you forward, negating any loss of speed from the brake feathering.
Trail braking was described to us as this - heading into a turn at a speed faster than will allow you to make the turn, you apply threshold braking in a straight line to srub as much speed as possible. When it comes time to enter the turn, you still have too much speed to make the corner. You're also using all available traction for braking, and there is nothing left for turning. So, as you turn the steering wheel, you start to let off the brake pedal. You're basically changing the ratio of braking:turning, from 100:0, to 99:1, 98:2, 97:3, etc all the way to 0:100. This allows you to carry the largest amount of speed into and through tight corners, while still being able to slow yourself just enough to retain traction. You start with full braking, and slowly trail off the brake pressure, thus the name trail braking.
Both of these techniques were done on gravel. I don't know if that changes anything when on tarmac. I also fully admit that my understanding of the techniques may be wrong!
