Public Finance was always dealing with intervention through fiscal instruments, including federal relationships between different tiers of governments and later on it subsumed Public Finances dealing with public sector enterprises when many private enterprises were nationalized and/or new enterprises were established in the public sector. Although public works were very much in the public sector domain even if many of them were erected through outsourcing to contractors. Public Finance generally did not cover the regulatory aspects of intervention, except in the context of public utilities or natural monopolies. This becomes part of Public Economics. The process of democratic decision-making and problems associated therewith is now discussed under the rubric of public choice, which has been included by most economists in the scope of Public Economics.
Some textbooks are more concerned with evaluation and appraisal techniques of large projects while others dwell on the theory of second best as in real economies conditions of perfect competition barely exist and distortions in one part of the economy may have to be balanced by causing distortions elsewhere if the former cannot be done away with.
In short, study of Public Finance (including Public Revenue, Public Expenditure and Public Debt), Public Enterprises/Firms, Public Projects, Public Utilities, Public Services and Public Choice would broadly form the realm of Public Economics. They may be concerned with any level of government- national, sub-national or local.
Over last 50 years there has been a curious development in the matter of local governance. It has been argued that people may vote with feet rather than by show of hands. It means that individuals may quietly mere across local jurisdictions for the preferred package of local public goods like sanitation, road conditions, aesthetic view. Conditions of such phenomena/occurrence as this have been a matter of research.
However, we may note, public intervention through money supply, interest rate, foreign exchange rate, etc. by monetary authorities is still not covered by most textbooks in its domain though it logically belongs to Public Economics. Part of the reason for non-inclusion of this type of public intervention is the fact that most of Public Economics grew out of microeconomic theory as applied to public intervention in market whereas most of monetary and exchange rate intervention comes from macroeconomic domain.