See my earlier reply for what is I think a workable definition or let’s discuss why that is not a good definition.

For how things are different per country OSM_tags_for_routing/Access_restrictions is likely a good source. For the UK, the table shows no use of bicycle=designated following the answer of @Richard.

The Default access restrictions for all countries unless otherwise specified on that page has bicycle=designated and checking all countries I see most countries follow that except for a few.