What bicycle=designated means to me is that the path/way is specifically meant for cyclists, the cyclist is the primary user.
A highway=cycleway has already bicycle=designated as default default access restriction so no need to add it. A footway with “cycling allowed” is bicycle=yes.
The only other ways that fit this are a cycle street and (for the Netherlands) a path that has a red surface but no sign although that will almost always be mapped as highway=cycleway, not as highway=path + bicycle=designated.
Also two adjacent paths without a sign are assumed to be highway=cycleway + highway=footway but the footway is almost never separately mapped, instead foot=yes and sometimes sidewalk=right are added to the cycleway
The type of cycleway is mapped with traffic_sign=NL:G11/NL:G12a/NL:G13:
verplicht = compulsory, a G13 is not compulsory
sideways are not allowed for cycling AFAIK and on a path that is not signed as footway it is not forbidden to cycle but not too many people know that and it is often not done. These paths are also often mapped as highway=footway while strictly speaking highway=path would make sense.