Historically there have been differences between CS and FAR, but recent efforts have been made to harmonise these regulations (AC 29).
Minima for low visibility operations are (have been) a combination of the aircraft equipment fit and certification, the operational requirements including visibility, and the standard of ground aids / lighting.
The altitude element can be related to the installed equipment and aircraft performance, from which the visibility follows and usually has to consider cockpit cut off angle.
JAR-OPS 1 attempted to rationalise many different European operational interpretations, including those based on the valuable ECAC doc 17 and the UK CAA fog model. The resultant regulations, IMHO, are a collection of the lowest standards which everyone can now subscribe to without much consideration of some of the hazards, e.g. auto Cat2, DH 100, is allowed to continue to 80ft in reduced visibility, but the new regulation fails to address what action to take if the autos fail between 100ft and 80ft – the visibility now being less than that previously agreed for manual landing.
Alternative, successful fuzziness in the regulations is seen in the European Super Fail Passive operation – Cat IIIa 50ft DH, in 150m RVR.
Then there are debates about ‘see to land’ (auto / manual), and ‘see to verify’ (auto Cat IIIb AH).