A lane would be a space on the carriageway reserved for bicycles.
A path does not necessarily need to be on a carriageway, but could include paved or unpaved areas reserved for bicycles. Because mountain bikes use paths also, often these are just dirt paths.
A route could include both paths and lanes. (And is usually designated by the city, town, or county authorities.)
A track would most likely be a paved oval or circular area that bicycles could use for racing or for training.
I think that none of these is specific to in or out of town, but cities are most likely to provide them, whereas rural areas at best would have cycle paths or cycle lanes.
Cycle lane - a lane reserved for bicycles on a road
Cycle path - a path specially provided for bicycles, with a metalled surface
Cycle track - a circuit designed for racing or racing training.
Cycle route - a route for cyclists defined by an authority - a local council, a national cycling body, etc. The cycle route may include a mixture of normal roads, cycle lanes and cycle paths