Ology = study of, in latin and herp = reptile/amphibian, in latin, so natural conclusion is that a herpatologist studies reptiles and or amphibians. They work for who ever wants research done on these animals .
It depends on who hires you and for what. Museums hire some for each of the job descriptions you mentioned plus others. National parks and wildlife hire others to do field work, each state authority on animal research and protection also hire herpetologists.
Asking what a herpatologist does is like asking what food people eat. It is too varied and subjective to the individual. And if you knew that why did you ask?