Q.: Which is the correct term: ‘European Union’, ‘European Community’ or ‘European Commission’?
A.:They are all correct for different occasions.

The European Union came into being with the entry into force of the Maastricht Treaty on 1 November 1993. In discussing events prior to this date the term ‘European Community’ or even its predecessor, "European Economic Community", would be more accurate. Since 1 November 1993, the term ‘European Union’ has replaced in common usage the term ‘European Community’ which in most cases refers to the group of Member States making up the European Union.

The European Commission on the other hand, is the executive body of the European Union. The European Union, as such, has no overseas diplomatic representation. It is the European Commission that is represented by diplomatic missions overseas and the correct appellation, for example, is: Delegation of the European Commission to Indonesia, Brunei Darussalam and East Timor.
Source: Wikipedia