As mentioned by others it maybe rather a business question. But even for native speakers the term öffentlich may be misunderstood, because in Germany there is often talk of öffentliche Hand and öffentlicher Dienst.
It seems to me, öffentlich as a term is in use for two cases here (perhaps also carelessly translated and used):
public accounting => öffentlich dargelegte Rechnungslegung - see also Public vs Private Accounting
- in the sense of making it public. Public accounting involves being done by an as possible independent that examines the financial statements e.g. by following rules to see if those fairly represent the results.
public accounting => öffentliche Rechnungslegung
- As mentioned by you referring to the accounting standards of (German) governmental institutions, also known as cameralistics. But please note the transformation of the public budget and accounting system from cameralistic to double-entry accounting style ("Doppik") is under implementation.
The "Externes Rechnungswesen" (or also used: .. Rechnungslegung) reflects the financial situation of the company to the outside world. The addressees are investors, creditors, the tax office, customers and suppliers, the State and also the interested public.
This term term should not used in the sense you mentioned above.
Controlling (derived from to control, 'steuern', 'regeln') is a term used in economics and is understood in the German-speaking world as a sub-function of the corporate management system (management), whose core task is the planning, management and control of all corporate divisions. A direct equivalent of the German language is unknown in the English-speaking world (reference: German Wikipedia).
So, Controlling is not a synonym of internes Rechnungswesen.
You maybe interested having a look at the Mindmap. Please note that this (arrows) is not a representation of the work flows.