I think your tagging is about right.
Though is it actually a driveway? I would usually only tag that if its just a road to a single house. Whereas that looks likes like it might go to several different buildings / houses or a factory? So maybe just tag it as highway=service, without specifying the type of service road. I think this will improve the rendering a bit, so it won’t disappear before the path.
As for Mapnik, showing designation, AFAIK not much has happened with that. The Mapnik layer is a rather general purpose worldwide map, so its probably not going to show things specific to rights of ways in England and Wales.
If you’ve not seen it already, worth a look at Freemap. It is a map specifically designed for walking in the UK, so should highlight this sort of thing: http://free-map.org.uk/freemap/
Or you could try making your own maps to highlight whatever you want.