To answer the first part a LSD takes power away fromo one wheel if it begins to lose traction and puts it to other wheel. Think about doing a burnout. If you have enough power you can sit there spinning one wheel withough LSD. With LSD it sends some power to the other wheel allowiing more traction and a quicker start. As for the HLSD and VLSD, i am not sure. There are a few different types of LSD's.
A differential by definition allows both wheels to turn at different speeds. This is because both driven wheels follow a different radious arc in a turn - the inside wheel must turn less than the outside wheel. But, an "open" differential will transmit the most torque to the wheel with the least resistance.
A limited slip differential (LSD) by definition limits the differential action. There are several different types of LSDs. Some are viscous (our stock units), some are gear type (Torsen - aka ATB type - the Quiafe is an ATB or automatic torque biasing), and some are clutch types. There are others as well, but I'm less familiar with them and won't go into them here as they are not available for our cars.
Anyway, all LSDs have some method of limiting the difference in rotational speed of both axles.
If you have a Quaife/Stock Spec V then both tires will spin at once in low traction situations. This could be interpreted as a push. The stock B13/14 LSD will allow one wheel to slip for a little while before both start to spin. This could be interpreted as less of a push but still a push. An open diff on a fwd car will usually let one wheel track around a corner and not spin. This would feel normal to most people. If you are not used to an LSD on a FWD car then this might be the issue you are describing.