I'm not sure I'm satisfied by that definition of corruption; a more coherent definition might be the use of authority against the consent or interests of those subject to that authority.

The reason is that corrupt acts or policies might cause no private gain, or may even cause public gain. For instance of the former, an authority may create economic policies known to worsen economic conditions for the whole of the population it governs, including those in authority. For instance of the latter, an authority may engage in a war of conquest against a foreign population, but share the spoils with the population it governs.

I think both are clear cases of corruption, but not necessarily with clear private advantage.

It depends whether, by "government", you mean "state" or you mean "system of governance".

When most people refer to government (of a nation or country, rather than something like student government of a university), they are referring to a state. A state is defined as an institution which has, and either wields or reserves the right and capacity to wield, a monopoly on the legitimate use of force (violence).

Some nations or countries may lack a state (like Somalia, or the Duwamish) and may be ruled or occupied by another state (like Palestine or Tibet). In some of these cases, such as Palestine, some form of governance may exist (in order to delegate rote tasks like refuse collection), while in others, many (or no) such system may be in place.

As another example, during the Spanish Civil War, parts of Spain had been removed mostly or entirely from the aegis of any state's power, but operated under formal governance without an institution that claimed a monopoly on the legitimate use of force.

All of these examples are very different, and help to illustrate my answer:

Absent a state, corruption can be eliminated (anarchist Spain tended in this direction before its successful destruction by communist and fascist forces; that isn't to say that it was a utopia, but that its trajectory was toward the creation of a truly egalitarian society) but may not be (Somalia, Palestine).

If a state exists, corruption is not just inevitable but built into governance by design. A claim to have monopoly on the legitimate use of force is itself a corruption, by definition. Such an institution grants itself, under threat of violence, the ability and even the duty to define (and redefine) virtue or right behavior as it sees fit, and obligates itself to treat as illegitimate and repress any challenge to its determinations in this regard (failure to do so undermines its monopoly, and thus its legitimacy).

The corruption of a state can vary, depending both on its strategy to maintain power (the more democratic states tend to favor "soft power" where effective; more autocratic states tend to be more repressive) and on its fragility (a contested state tends to wield its monopoly more), but at its root a state is inherently built on coercion. The only form of authority which is not tainted in this way is that which derives solely from the consent of its subjects. Consent and coercion being mutually incompatible, a state is inherently corrupt.

This may seem like a wordy regurgitation of the adage, "power corrupts...", but it should be taken to be more nuanced. Power is not what's being denounced: a talented athlete, for example, can possess and exercise power without resorting to coercion. As can a talented orator. Power should be understood as the ability to effect, not necessarily to coerce.