At your friendly neighbourhood airline we have such a conversion table in the "Quick Reference Pages", so easily looked up in flight. This table has 11 possible ILS classifications (from I/A/1 to III/E/4). The CAT III section alone has 8 sub-classifications. Some allowing operations down to CAT I minima, some to CAT II and others to CAT III minima. I know that doesn't sound logical, but while the "Facility Performance" (the first character) may be CAT III, the "Level of Integrity" (the third character) may restrict ops to e.g. CAT I. The second character, by the way, denotes the "Limit of Course Structure", whatever that may be.
Even when CAT III ops are authorized, distinctions are made between CAT IIIB with DH and without DH, and for the RVR limits to be used.
Long story short: just telling pilots that "the ILS supports CAT III operations" is definitely not telling the whole story.
And re. pragmatic: works fine for us!