@anonxanemone , I would only add to your list specifically Graphically Unique Structures, whether Buildings, Improvements or Districts.
Your theme of a Unique Culture being absent from the game's Civs has been touched on in other threads, usually in the context of some kind of Underlying Population in every tile that either contributes to or has to be converted to your Civ, or a combination of many factors in the game now as a specific 'Culture' unique to each Civ.
Right now, many of the components of Culture are in fact chosen by the Player: Religion and all of its beliefs and tenets, Social and Civic 'Policies', types of Government, etc. Only Language is not addressed specifically, but one can assume that as part of the 'basic makeup' of each individual in-game Civ.
This is not a bad system, because elements of 'Culture' do change (usually slowly and haltingly) throughout history, and having a 'standard culture' for each Civ wold have to assume the development of that Civ will follow its historical model in every game, a concept which, frankly, would make me wonder why I should bother playing the game in the first place.
What the current system leaves out is that many aspects of culture, including Religion and Social 'Policies' are not chosen by design, but emerge as a result of events and mingling of populations - in other words, what happens in the individual game you are playing, not necessarily what you Want to happen. It also leaves out Multi-Cultural Civs, which have been around since the Beginning: the early Mesopotamian City-States and 'Empires' were all composed of individual cities that in many cases had developed somewhat different cultures, and in some cases languages, and today to regard, say, the United States or Great Britain or even China as having only One Culture Each would be very hard to justify.
I, for one, would like to see Culture as something more than just a set of Points that convert to Tourism that converts to a 'Cultural Victory'. It should be a defining characteristic or set of characteristics of the Civ, subject to change, resulting in occasional conflict or even major disruption . . .