In my opinion, there is this group of parliaments which consists of Congress, National Assemblies, Duma, Parliament, Diet, Knesset and whatever name can country call their parliament (it basically is the representative body of people formed by elections - at least one chamber of parliament).
It doesn't really matter if the system is parliamentary or presidential (even semi-presidential or that special one they have in Switzerland). In all of the countries, this body has a legislative function (so it is called a legislature). Then you can distinguish the systems where even if you have parliament (Supreme People's Assembly in DPRK) it doesn't really have legislative function (or such function is just nominal).
Heads of state and heads of government can also be pretty diverse, like you mentioned presidents, monarchs, prime ministers and so on. Sometimes is HoS and HoG the same - like in USA or Switzerland, sometimes not like in Sweden, UK or Germany (where they have HoG as Prime ministers or Chancellor for Germany). There are even countries like Russia or France where president and prime minister coexist, but President is the HoS and HoG, and he has the upper hand in executive and representative matters.
Whole another thing is electoral system in country. You can have proportional, plurality or mixed electoral system (and a few more, but they are pretty obscure) in both presidential and parliamentary government systems. There you have plurality elections for parliament in presidential (USA) or in parliamentary system (UK for example). Also you can have proportional elections for parliament in presidential system (Costa Rica) or in parliamentary (Austria).