It’s far and away best to map the track as highway=track with the designation=public_footpath. It is one of the things which makes a difference when using the data. The main map is not really optimised for walking in the UK, changes made years ago made it less useful that it had been before. Fortunately SomeoneElse, who is a very keen walker, maintains his personal map for Britain and Ireland which is optimised in this way. This clearly shows different types of rights of way, walking routes, LDPs, stiles and other footpath furniture, field boundaries etc etc. This is the area of your PRoW: https://map.atownsend.org.uk/maps/map/map.html#zoom=15&lat=51.2777&lon=0.05201. The public footpath is shown in red with longer dashes for when it’s on a track and short dots when its a footpath.
I have noticed quite a lot of “tracks” with incorrect access. Eg. Gravel tracks down to farms, that on foldable paper maps commonly available are marked Public Footpaths and have been entered as Unmaintained Track Road, designation Public Footpath, BUT with bicycle=yes, horse=designated. The farmer allows this for people he knows, but shouldn’t bike/horse be left Unspecified or possibly Private?
Yes, use of bicycle=designated is definitely wrong. The only guaranteed accesses are foot=yes (because it’s a public footpath) and access=destination (to allow posties, delivery drivers, etc to visit the farm). I suspect if only certain people known to the owner are allowed it’s still generally private usage rather than permissive usage. It’s been noticeable how much people are using OSM to find local paths and this has shown that certain routes were either mapped over enthusiastically or people didnt check whether presence of a path meant that it was a public one.
In some cases highway=service with service=driveway may be more appropriate than track (particularly if larger vehicles use it). However this is a nuance.
I sometimes see “bicycle=yes” added to public footpaths on driveways in error because someone cycled down it once despite there being no widely accepted legal right to cycle there (I say “widely accepted” because I know that people have tried to make the case that a bicycle is a "usual accompaniment’ in a legal sense of someone walking along the path - for discussion see http://www.peakandnorthern.org.uk/newsletter/1309-allowed.htm and elsewhere, but for now let’s assume that cyclists don’t have a legal right to cycle somewhere just because it is a public footpath).
Also, some public footpaths are routed along constructions designed to allow cycling - https://www.openstreetmap.org/way/818917531 has been built as a cycleway (previously it was a forest track) but is designated as a public footpath.