The chief minister for any government ministry or department is the Secretary of State for [whatever]. Under him or her there are Ministers of State, and at the most junior level there are Parliamentary Under-Secretaries of State.
The exception is HM Treasury as it is the oldest and most important department (it controls the money for all the rest!) The equivalent of the Secretary of State is the Chancellor of the Exchequer, and all the junior ministers are [something] Secretary to the Treasury. So there is no "Minister of Finance" - where did you get that idea from?
They are all ministers. But their proper titles are long so the media usually shortens them in some way. Thus they will write Education Secretary instead of Secretary of State for Education. A Minister of State in the same department might be called an education minister, and Parliamentary Under-Secretaries of State are often referred to as junior ministers. The Chief Secretary to the Treasury is often called just the Chief Secretary as it can't possibly mean anyone else, and the Chancellor of the Exchequer is usually referred to as just the Chancellor as there is only one minister with Chancellor in the name - apart from the Lord Chancellor, and that is always written out in full as it's only two words. Not that it is often needed anyway as since a recent reform, the Lord Chancellor is always also the Secretary of State for Justice.