What type of physicist works in the field of Classical Mechanics? And yes, this does encompass the general theory of relativity.
What type of physicist works in the field of Classical Mechanics?
It depends on what you mean by "classical mechanics." Does this term encompass general relativity? General relativity is after all a classical (i.e., non-quantum) theory.
If not, the answer is for the most part they don't. The only exception might be physicists who work in the field of fluid dynamics. Otherwise, classical mechanics is now in the domain of engineering.
Fluid dynamics gathers interests from many corners. People who study fluid dynamics include engineers (many disciplines), but also chemists, biologists, meteorologists, oceanographers, geologists, mathematicians, ...But someone doing theoretical research in fluid mechanics would be regarded as an engineer.... or?
There are many good answers here, but just to add another:
Plasma physics is a surprising rich field that is entirely classical. Closely related to fluid dynamics.