In Barbados, which is a parliamentary republic as a result of Elizabeth dying, they had a system where the PM and Opposition Leader could agree on who should be nominated. If they can, then the speaker asks the MPs if they object. If anyone does, then they hold a contested vote. If the PM and leader of the opposition can't agree, then they have a contested vote. They vote until 2/3 of each house agree on who it should be. They can be dismissed from the job by a tribunal formed specifically to judge them based on misconduct.
Ireland directly elects them, and is also a Westminster system as Canada is.
Edit: I for some reason can't reply directly and so I have to address them here. Germany and Italy do not have their chancellors or prime ministers choosing a president. That is the exclusive role of the electoral college in both countries, in Germany consisting of the Bhndestag and an equal Number of delegates chosen by the state parliaments, and in Italy it is the members of both houses and some regional council presidents. The electors vote by secret ballot with runoff ballots if needed. I don't know how this could have been interpreted as prime minister and parliament.