CAT II/III minima are based on various requirements as well as ground equipment.
One being that Touch down zone lights are required for CAT III ops and not for CAT II.
Another is that center line lights are required every 15m for CAT III and every 30m for CAT II ops.
With that being said WK, the airport you landed at may have had the equipment that satisfy CAT II operations and not CAT III. Refer to the 10-1 or 10-9a pages of the Jeppys. Also, you Ops Specs may limit your operations to minima greater than CAT III regardless what is contained in the Jeppys. RA minima are set as appropriate.