There are only two sub-fields of economics on your list. Public Economics and Political Economy.
Public economics is the field of economics that studies government's role in the economy. As an example, what are the welfare effects of taxes, subsidies, and economic legislation?
Political economy is the study of the interaction between the economy and legal and social institutions. As an example, does democracy affect the economy and/or does economic inequality affect democracy?
Economic policy, to me, is the set of policy prescriptions that comes from economic research (Research shows X is associated with Y, if you like Y policy should do more X). It is something that can be an outcome of any field - labor, health, and so on. Public Policy is more of an art, it's the process of how ideas (such as, but not limited to economic policy suggestions) actually become policy. Degree programs in public policy study that process from both theoretical and practical viewpoints.