One is a unit linked to government functions, with a company name and a legal person, basically belonging to the institution.
The second is the third-party organization you said, belonging to a private enterprise or joint-stock company, the government trusteeship of the sewage treatment plant business is entrusted to such an organization. This kind of trusteeship is generally related to the construction project when the BOT model is used, it is also possible that the third-party organization directly from the government to acquire the sewage treatment plant.