The term "public service" is rather vague and can have different meanings in different contexts, but no matter which meaning we use, it is markedly different from public cooperation.
In the context of businesses and individuals, there are a few ways to view this term. When we speak of a company performing a public service, we may mean that the company is doing something for the benefit of the community. For example, when I was a little girl, there was an enormous ice cream factory near my house, and the company owned a large lot of land upon which it put benches and a playground for children, generally beautifying the lot with trees and flowers and so on. Everyone could use this area, whether stopping for ice cream or not. This was one kind of public service. An individual can perform a public service, too. If a major crime has been committed and the police are unable to solve it, an individual can post a reward for information leading to a conviction. This is a public service. We also speak of public service in the sense that a service is open to the public. So, a supermarket is a form of public service, while a private club is not.
In government, an individual who is working for the government is said to be in the public service sector. This would include teachers, politicians, and the local dogcatcher. We speak of government as providing public services, which can include housing, education, or a municipal swimming pool.
Public cooperation is the voluntary cooperation of members of the public, generally, as part of the social contract, or for a specific reason. If we did not all stop when the light turned red, there would not be enough policemen to catch all of us. This is part of the social contract, and it could reasonably be said that this is a form of public cooperation. But sometimes we are called upon to cooperate under specific circumstances. In California, as the drought has gone on for year after year, the governor has called upon the citizenry to cooperate in water conservation. When two dangerous men escaped from a prison in upstate New York, the public was called upon to stay alert. When hurricanes are imminent, people are asked to cooperate in any number of ways, by evacuating in an orderly way or by helping to pile up sandbags. These are all public cooperation.
You can see that these are very different concepts, no matter what definition you apply to public service. In one case, businesses, individuals, or government provide a service, and in the other, the public cooperates for the good of all, generally, or in specific situations.