A path indication on the map means that when the mapper mapped it there was physically a path on the ground. It gives no indication as to who is or is not allowed to use that path.
The brown background indicates "access land". You can walk on this land subject to some restrictions. This applies regardless of whether there happens to be a path. "access land" is a relatively recent concept. It does not in of itself give any permission to cycle or ride horses.
The green markings indicate public rights of way. A public footpath is somewhere that people are legally allowed to walk even if the landowner doesn't much like the idea.
A public bridleway similarly allows the public to walk, cycle or ride horses.
A restricted byway aditionally allows non-motorised vehicles and a full byway allows all traffic.
Often if you look carefully you will see that the green "public right of way" markings are printed on top of a marking indicating what (path/track/etc) is actually on the ground. Sometimes the official route of the right of way doesn't exactly line up with the actual path. You can see this on your map where there is a "kink" in the public right of way but the actual path underneath goes straight through.
Unfortunately while public footpaths, bridleways and byways are explicitly marked on OS maps there seems to be no disctinction on the map between public and private roads.
You also get "permissive footpaths" where the landowner has decided to allow people to walk and permissive bridleways where the landowner has decided to allow people to walk/cycle/horseride. These are indicated in orange on the OS maps. The importand difference from public footpaths/bridleways is that the landowner can revoke the permission.
Note that the above applies to England and Wales. In Scotland public rights of way are not shown on OS maps because there is no definitive map of them, only historic convention.