Hi, 

There's nothing on OpenStreetMap which says anything else. A track without any other information is always just as you say a private farm track. This is exactly the same as with Ordnance Survey maps, a path, road or track does not imply any rights of access. This map of the same area explicitly shows which paths & tracks are public rights of way (in colour) and those which either are not or no information is available (black): https://map.atownsend.org.uk/maps/map/map.html#zoom=18&lat=51.350577&lon=-0.250476

Unfortunately people do seem to feel entitled to use things just because they've seen them on a map. Also some people are using apps which originate in other countries where farm tracks may be generally accessible. 

If you have further problems, I'd encourage you to ask people if they are using an app & which one. This helps us (OSM contributors) to try & resolve things generally with any app developers/vendors.

In the meantime you can explicitly add access conditions to tracks & paths using the OSM editor, which may alleviate the problem. This is as simple as opening the editor clicking on the track & adding private in a field on the form on the LHS.

Hope this is of some help.

Jerry aka SK53

PS. One minor point is that service roads & tracks providing access to farms may need destination rather than private because OSM is now heavily used by parcel firms, including Amazon.