My 2cts:

  1. Is the way made for / designated to bicycles?
    a. Yes: then highway=cycleway.
    Proof is ideally a traffic sign, but other markings or typical design (defined locally) may tell the tale.
    bicycle=yes is implied. All other legal and customary implications or exclusions differ by country / region / maybe even by city. Probably foot=yes can safely be assumed, and horse=no, but vehicle access differs widely AND changes over time. No surface type should be assumed. So many situations require explicit tagging.

b. No; it’s not a cycleway. Goto 2.

2 Is this a type of way where bicycles normally may ride AND where bicycles do not have an alternative they are supposed to use?
This is country-specific. Use the access-wiki with access rules by country and by road type.
Yes: you’re fine.

No: if it’s an exception. tag bicycle=yes.
Not clear: tag bicycle=yes to be sure.

  1. Is this a type of way where bicycles normally may ride, but has been dedicated firstly to cycling, while still allowing other vehicles “as guests” with regard to speed and priority?
    tag bicycle=designated and explicit access for all other vehicles. There is probably a speed limit as well.

If bicycles have an alternative they are supposed to use (e.g. a lane or a side path) tag it on the highway (e.g. with bicycle=lane or bicycle=use_sidepath)

In all non-standard situations, tag explicit access. E.g. if a moped is allowed on the separately mapped cycleway as well as on the road, tag moped=yes on the road AND on the cycleway.

In short,
bicycle=yes means access for bicycles.
highway=cycleway means it’s a cycleway verifiably (signs, markings, construction) made for or assigned to bicycles; bicycle=designated is implied, but other access rules (exclusions/inclusions) may vary.
bicycle=designated on a different road type or path means it’s not a cycleway per se but it’s formally dedicated firstly to bicycles, while not implicitly excluding other traffic.

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