This could be kept rather simple. Your assumptions are correct, RAIM is a self-evaluating algorithm internal to the on-board box, clue the name.
The receiver needs signals from 4 satellites to calculate its position in 3D space. If 5 were in view it is possible to calculate 5 separate, semi-independent positions, by excluding one satellite and using the remaining 4 for triangulation. Comparing those 5 positions one against another is what RAIM does and if a large enough discrepancy is found a fault is flagged.
GPS RAIM not available simply means there will not be 5 satellites in view. More rules to the game, but such is the layout of the course.
For the last sentence:
- you need RAIM to fly the approach
- if additional corrective signals (augmentation) are available, it is allowed to use lower minima
on that RAIM-required approach since finer precision is assumed.
Approaches without RAIM do not exist, just like it is impossible to fly an ILS without somebody monitoring the signal and checking it on regular basis.
An indicated position without means to
- ascertain the signal is valid
- quantify the magnitude of possible location error
is useless and never used(*).
(it is on Air Crash investigators
)