I guess that in your second category indirect parliamentarism the most common arrangement is actually for the Head of State monarch or president to appoint a prime minister and then appoint all other government of ministers on proposal of the latter That is what happens both in the UK where the power to appoint the government rests with the Queen  and in the France where it is the President of the Republic who appoints the Council of Ministers In Germany the Federal President also appoints all other federal ministers on the recommendation of the chancellor but the chancellor himherself is actually elected by the Bundestag thus making Germany fall into your proposed direct parliamentarism category instead