In India, what is the difference between "local flying area" and "airspace"?
A local flying area is airspace that is designated by local air traffic control for general flying not involving circuits and landing. In the US it is something that is loosely referred to as "the practice area." In this area aircraft can perform practice and other maneuvers, and air traffic control will be aware of it and can warn other aircraft to be on the lookout for this. In India it is common for these practice areas to be segregated for flying school aircraft, and medium to heavy aircraft.

To put it another way, all local flying areas are part of a country's airspace, but only a fraction of the airspace is designated as local flying areas. I hope this helps.