Please excuse my pedantry, but you've misspoken here: the European Council, the Council of the European Union, and the Council of Europe are three different things; and here you've accidentally said the third when you meant to speak of the second. The first two are EU institutions–the European Council is senior to the Council of the European Union, in the former EU member states are represented by heads of state or government, in the later they are represented by the ministers responsible for various policy areas (finance, agriculture, trade, transport, education, etc). The Council of Europe, by contrast, is a separate international organisation from the EU, with a much broader membership – it includes the vast majority of European countries; it is most famous for hosting the European Court of Human Rights (ECHR), although it is active in other policy areas as well.
And all three have a presidency. The President of the European Council is an individual, currently former Belgian Prime Minister Charles Michel. He doesn't take orders from the Belgian government, he is truly independent. By contrast, the presidencies of the Council of the European Union, and the Council of Europe, are held by governments not by individuals. The presidency of the Council of the European Union rotates every six months, it is currently held by Portugal, Slovenia is up next. Likewise, the presidency of the Council of Europe rotates every six months; currently it is held by Hungary.