Personally I’d be tempted to tag the links with ncn=link, ncn_ref=(66) - or the equivalent with relations.
LCN tagging in OSM is, however, a mess. Best practice IMO is to use it for routes signposted to somewhere.
For “paved paths divided into a pedestrian half and a cycle half”, I’d use highway=cycleway, segregated=yes, surface=asphalt. For the undivided path, highway=cycleway, surface=asphalt. You don’t need to tag foot=yes, highway=cycleway implies that.
The defintion of highway=track says “Roads for agricultural use, gravel roads in the forest etc”. So if it appears to be used by farm tractors etc, I’d tag it as a track.
If its only used by pedestrians / cyclists / horses, or not wide enough to fit a car / tractor etc along, then I’d tag it as a cycleway (or a footway / path etc).
And if its a track, I’d say its worth using surface and tracktype tags, just to make it clear. Though I suppose tracktype=grade1 probably implies surface=paved.
Also worth using bicycle=yes, just to make it clear. Its also helpful for routing software, which may not check all of the route relations.
The signs on the ground are muddled which probably accounts for the confusion on the map. Is this common? The route that is labelled as (66) on opencyclemap would seem to be the 66 itself, and not “the 66 is over there” at all. The blue “local” signposts all show something like “University (66)” but the stick on labels on posts etc all show the 66 without brackets.
The fact that there’s a milepost on this track suggests it’s the “real” 66 as well.
OCM currently shows the 66 heading west along a disused railway and then turning north, but this appears to be incorrect, it does actually head south to the Uni to join the “(66)” and ultimately the 65 just south of the centre. The disused railway is a local track into York, but there is one (only) “stray” 66 stick-on sign on this path too!
What a mess!
I’ve checked on the Sustrans map and it is clear where the 66 goes so I will put it all right as I’ve cycled it and have confirmed the route.
Sustrans is attempting to bring some order to NCN numbering by renumbering “braids” and Regional Routes as (usually three-digit) National Routes. But there’s still quite a lot of places where the numbering is confused. Your local Rangers may be able to supply enlightenment.