In the States, runways are certified as either Category III or not. The aircraft systems and operator programs are certified as either fail passive or fail operational, and are limited to some point on a ladder of decreasing RVR values as the program develops. While there is a definition of CAT IIIA and IIIB still in the advisory material, the only people really still using this terminology are the Jeppesen folks...over here, at any rate.
The only technical reason for a DH, on any approach, is the requirement to use visual references. For CAT II approaches, this means that the lighting must be adequate to manually maneuver the aircraft into the touchdown zone. I realize that few operators allow manual CAT II landings anymore, but that is what the visual references are predicated on. Note that in the States, we now have waivers allowing CAT II approaches with diminished lighting as long as we autoland...which sort of restates the preceding point.
For CAT III fail passive approaches, the visual references are only necessary to confirm the aircraft's position in space at 50 feet. This provides the redundancy required for an autolanding that is otherwise unavailable because you are only using two autopilots, or two channels of one autopilot. Essentially, you need to see green TDZ lights on the left and green TDZ lights on the right.
I have no clue why the French require a visual reference at 14 feet; I see that Airbus recognizes this in the document of theirs I found on Skybrary. They say for such low DHs, you only need to see one centerline light. Presumably it needs to be right in front of you, and hopefully it is a centerline light!
However, silly as it may seem, one would have to investigate the construction and maintenance of their ground-based systems, i.e., localizer, system monitors, protections from interference, etc..before drawing any conclusions. The FAA AC 120-28D defines it thus:
A Decision Height is applied to all Fail Passive operations and is specified at certain locations where fail operational minima is authorized. For Category III, a Decision Height is usually based on a specified radio altitude above terrain on the final approach or touchdown zone. The Decision Height is established to assure that prior to passing that point the pilot is able to determine that adequate visual reference exists to allow verification that the aircraft should touch down in the touchdown zone.