The term representative democracy is principally used to distinguish it from participatory (or direct) democracy, in which citizens directly debate and vote on legislation, policy and the appointment of citizens to execute those policies.

Both forms of democracy are common, worldwide. The main advantage of a presidential system is that it yields a strong and clear executive. This is important, especially in times of crisis or war. In contrast, parliamentary systems often result in a fragile or even absent executive.

In a democracy, politicians over time gain experience of the areas that interest them, policy and policy-making and they learn to work with different branches of government. More, in a parliamentary democracy, they learn to gain the respect and the trust of their constituents by engaging with them in their local offices.

I would argue that there is no one, perfect, overarching explanation for why every authoritarian figure rises to power in a multi-party parliamentary system. The history and context of individual political systems, societies, and nations can have different effects and consequences.

The definition of a Parliamentary Democracy is one in which an elected Parliament is Sovereign. It doesn't require a Prime Minister, nor is the definition dependent on whether a Prime Minister must have an explicit vote of confidence or an implicit vote of confidence at the start of their appointment.