Precision vertical guidance is not really based on altitude, it is based on your being at a certain point in space.

When doing GPS vertical guidance, the GPS is used to guide the airplane laterally and vertically so as to arrive at the correct point at the right time. The aircraft altimeter is used for situational awareness and minimums, just like with an ILS GS.

A certain number of satellites(3-4) are required to get a fix, plus one additional satellite for RAIM. Some GPS installations use baro-aiding, where data from the aircraft's altimeter is fed to the GPS box for use as an additional fix plane. This can allow the use of one less satellite without losing RAIM.

There are many errors which can affect the accuracy of the GPS signal, and these vary over time. The WAAS system corrects for some of this by sampling the signal at various fixed ground stations (where the actual position is known for certain), determining the error, and then transmitting the correction via satellite to the user receivers which automatically apply the correction.

WAAS has a limitation, in that it uses only a handful of ground stations, so the error at the nearest ground station is probably not the same as the location where you want to shoot an approach.

LAAS will solve this problem by sampling the GPS signal at or near each airport, and provide a custom error correction for that location and time. This will allow lower GPS minimums in the future.