In the US, "government" refers to the entire apparatus running the state. It refers to the legislative, executive, and judicial branches. In the UK, it appears to refer to the executive:
The Government runs the country and has responsibility for developing and implementing policy and for drafting laws. It is also known as the Executive.
The head of the government is the Prime Minister, and power is then delegated to the cabinet. Since the cabinet is the highest level of authority other than the Prime Minister, it is often used interchangeably with "government".
When it comes to how the terms are used in Ukraine, I would expect it most of the discussions to take place in Ukrainian, and thus which words are used in an English language newspaper would be based on the usage of the person doing the translating. The article you linked to is an American company, so it is probably intends the US meanings.