I've now re-tagged all of the private tracks (Airfield Lane included) so they should now clearly appear as greyed-out on the map, and all routing software using OSM should no longer allow cutting through them (they were only tagged as motor_vehicle=private before, not outright private).

This should help at least somewhat; while I agree that the universal solution to avoid confusion would be to hide the private tracks from the map and tag the public footpath track as highway=footway, unfortunately that doesn't fit with the current instructed provisions (https://wiki.openstreetmap.org/wiki/Access_provisions_in_the_United_Kingdom).

Lots of different third-party platforms use OSM as a base and many of those applications have different textures and rendering styles etc so while some would clearly indicate the private status, others wouldn't. As said above routing should work on all platforms though.

Although I see where you're coming from on not showing the track at all, as a contributor I can't really change the terminology of OSM alone. If you wanted to challenge it you could drop a suggestion at https://community.openstreetmap.org/.

Good luck with sorting the trespassing issues out, sorry I can't help more but hopefully this'll make a slight difference.