When looking at this the first thing to look at is where the touchdown markings (aiming point) is on the runway. Annex 14 is a good reference.
The distance between the threshold and the touchdown markings is not the same everywhere. It is longer at runways available for large aircraft then runways only used by smaller aircraft.
This first fact shows that if the runway is designed for a B747 and a 3 degree approach, the touchdown markings will be such a distance past the threshold that the B747 following a 3 degree approach towards those markings will safely cross the threshold (let's say 50ft clearance).
So for the B747, the LDA is matched to the LDR and the 50ft point is at the threshold.
Now if you hop in your Citation and make the same 3 degree approach to the same touchdown markings, you will follow the same path. As you cross the threshold, you will be more than 50ft. This is not a problem since the runway is long enough for B747's but if you want to cross the threshold at 50ft then your aiming point will have to be closer to the threshold than the touchdown markings provided.
Note that I have said nothing about PAPI or ILS up to this point.
ILS CAT1 provides certified guidance down to a minimum of 200ft above the runway. Below that, obstacle clearance is not provided even if the aircraft follows the indicated glide slope.
PAPI indications provide both obstacle clearance when on slope and also ensure that the aircraft reach the threshold at the appropriate crossing height.
Refer to the above (non-PAPI) explanation for why threshold crossing heights vary.
Thre objective of PAPI is to provide obstacle clearance and safe threshold crossing height - to intentionally fly below the PAPI indication puts the aircraft at possible risk of hitting an obstacle or not crossing the threshold at the safe minimum height.
The effect of not following the PAPI is also that the approach is no longer a 3 degree approach to the published touchdown point.
Finally - why do we sometimes have PAPI on both sides of the runway?
The answer is that the extra set if there to provide some indication of lateral level in situations where this is not provided by other means. i.e. if the minima could be such that you can't use the crosbars of the approach lights for lateral level guidance. If there are two sets (one each side) then they have to provide the same indications.
So - PAPIs are there for the sole purpose of ensuring obstacle clearance and a safe threshold crossing height on the appropriate approach angle. Ignore them at your peril.
Finally - when the threshold is displaced, the PAPI may provide less than the normal threshold crossing height. However, the normal height will be provided at the start of the runway.
Regards,
DFC