The origin of public interest theory can be traced to Pigou. I think land use control is based on that theory. However, economists appear to be more interested in public choice theories in recent decades. That also has implications to land use control and might be the key to understanding corruption in land use control.
There are issues that should be addressed through land use controls. These issues have their basis in technological external diseconomies (and economies). Specifically, we are addressing pollution (such as air and water quality differentials), noise, and congestion. Assertions about public "interest" that are found without reference to these diseconomies should not be considered as serious.
As Feng Deng has explained, there is a line of knowledge about public interest deriving from economics. Summarised rather loosely, this holds that the public interest is served by well-functioning competitive markets, so long as externalities, such as those identified by Peter Colwell, are accommodated.
There is a separate line of scholarship deriving from public administration studies. I summarised this work in my own doctoral dissertation. I argued that there are different meanings of the term. I argued that the term is always contextual (depends upon culture and circumstances of the particular polity being studied) but is never meaningless and that there are solid foundations on which a government can formalise an understanding of the public interest at a point in time for a particular purpose.
Dealing with the nature of land use regulation (rather than the nature of public interest as above), a departmental paper that I wrote a few years ago may be helpful. This is in the public domain. All of the ex-English colonies, including Australia and many African countries, follow broadly similar concepts of land tenure.
The paper explains that land use regulation or development control is a regulatory imposition that constrains (limits, restricts or shapes) the property rights that titleholders would otherwise exercise.
An ethic of public interest can be applied by the government at any stage of the process, right from the initial allocation of land. If the land was alienated from public control many years ago, then land use regulation/development control is the most effective method of giving effect to the public interest as it is the most powerful way of limiting private entrepreneurial ambition. An ethic of public interest can derive from the professional training of the regulators, from the legislation, from policies endorsed by the leaders and by the norms and practices of the public officers that apply the land use regulation. Any one of these sources can become corrupted, but operatives in any one of these sources can make a difference by applying an ethic of public interest as best they can.