It would be too complicated to go into all major details of political systems - the distinction of the Parliamentary dynamics of first-past-the-post and proportional elections, mono/bicameral Parliaments, federal, confederal and centralised systems, of PMs and chancellors is ignored here, and so is the connection to the judiciary and prosecutors. I also mostly ignore if differences are just in names (Assembly/Parliament).

For the benefit of our American readers, let's get familiar with colors on the basis of my graph for the US political system - a pure Presidential system:

Something close to the US system was Polish elective monarchy, and, save for the electability of the monarch, some earlier periods of British constitutional monarchy. But the King/Queen lost executive power there to the collective of ministers, called the government in Europe, and only kept ceremonial and representative functions. The government is headed by the Prime Minister (PM), who, owing to his lack of popular mandate, in theory should be seen as only a coordinator of the government, rather than autonomous decisionmaker.

This modern form of parliamentary democracy in a constitutional monarchy is also practised in Spain, the Benelux states and Scandinavian countries. As A swedish kind of death notes, the King/Queen's impartiality is ensured by the fear that the people could decide to switch to a Republic if s/he is unpopular.

The separation of executive power and representative, ceremonial powers was seen as sensible in many republics. But how to pick the figurehead President? Parliament could do it. Proponents of a parliamentary democracy with parliament-elected figurehead President see two advantages. One is that, in theory, the President could be an impartial representative of 'everyone', not just a majority (in practice, parliamentary votes mostly decide with a narrow politicised majority, but elected Presidents do strive for impartiality), and the other that Presidents won't have the popular mandate to attempt to usurp the authority of the PM.

Such a system is used by Germany, Italy, Hungary and some others. It gets a bit further complicated in Germany, where there is a Chancellor in place of PM, and the Chancellor's Office acts as a government within the government, dublicating policy units, thus increasing the head of government's powers relative to the other ministers.

A system of parliamentary democracy with popularly elected figurehead President functions in Portugal, Slovakia and Finland:

When the popularly elected President gains executive powers, and the power to remove holders of other branches of power, we have a dual system of parliamentary democracy with President. There are two centres of executive power. When parliamentary majority and President are from the same political camp, the system resembles the US one, with the PM having reduced significance - but the PM is there to catch the blame before the President if they made policy mistakes.

If Parliament and President are from different camps ("Cohabitation"), then it is very unlike the US system, then it is closer to the previous parliamentary democracy with popularly elected figurehead President system, for then the PM and the government will gain upper hand on most policies. Note that the President doesn't have much room for not picking the Parliament's preference as candidate for PM: the Parliament can vote the 'wrong choice' down, and the power to call for new elections is not worth much shortly after the previous.

This system is used most famously in France, but also in Austria, Poland and Russia (though as yet without an example of cohabitation in the last).

Finally, in Switzerland, we have an example of a consensus parliamentary democracy combined with direct democracy and without figurehead. In Switzerland, people regularly vote in referendums on proposals of both laws and policies, and also elect a national assembly. The Bundesrat (=federal council) is the executive governing body elected by the assembly, which consists of candidates of all major parties.

For one year, the assembly elects one member of the council to lead the latter, a post called Bundespräsident (=federal president), though it is a position without any extra powers, in fact closer to the PM ideal than any PM in practice in other countries. The figurehead post lacks totally, and so do most ceremonies entrusted to them in other countries(!).

Sorry for eventual further imprecisions and errors.