Here's some info in the context of India :
In India , a person drawing his salary directly from taxpayer's money is called
Civil Servant on the other hand carries a Raj era legacy when the country was governed by the Indian Civil Services. Even though the ICS is defunct now, the term Civil Services is reserved for mid to high level bureaucrats and other government officials. For instance, the government conducts civil services exams for recruitment of mid to high level officials and Staff Selection examinations for lower levels in its hierarchy.
So Public Servant is the all-encompassing word for anyone working for the government. A legal definition of the word is here
Public Sector in India has a special meaning. It refers only to various business organisations run by government due to its soft socialist nature. These include a number of petroleum companies , mining companies , banks etc. Their day to day operation is not supposed to be funded by government and in most cases they compete directly with their private counterparts and pay dividends to government. A huge multitude of people working in them are called
Public Sector Units (PSU) employees