I am not an anthropologist or a sociologist, but I’ve been thinking about this question for a while. I don’t claim to have a definitive answer but I would like to offer my two cents here.
I would define civilization as simply a highly urbanized society with an organized distribution of resources (both material and human). How much urbanization for it to count might be arbitrary but if there are a lot of people who are making food and providing resources for another group of people who distribute the resources amongst themselves then that is civilization. This leaves open whether the means of producing food is primarily agricultural, silvicultural, aquacultural, or pastoral in nature. This definition also doesn’t require writing, pottery, irrigation, as well as social stratification beyond ‘people who distribute resources.’ These latter attributes are supplementary and secondary innovations to the end of resource production and distribution. But a civilization only needs heavy urbanization and a level of organization to manage that urbanization.
Under this definition, Mesopotamia, Egypt, Indus River Valley, China, Mesoamerica, and the Andeans would all be true civilizations, and indeed they are the ‘cradles of civilization.’ There would also be little room to argue against including populations like the Nok culture, Mississippian culture, Cucuteni-Trypillia culture, Los Millares and El Argar cultures, and Ancestral Puebloans as pristine civilizations in their own right, or at least ‘proto-civilizations’ (if you think they are more borderline cases). This would however exclude groups like the Huns and Māori, who although did form state-like confederate structures did not need to manage their populations beyond the tribal level, as they did not have intensive urbanization. This doesn’t mean their society was inherently ‘lesser’ and civilization is something ‘more advanced. It just means that civilization is a way of managing population and resources under different circumstances that require it. As long as they are urbanized and intensified to the point of having to organize the distribution of resources and means of producing those resources across multiple urban population groups then they are civilization.
Notice, however, that I did not say ‘a civilization.’ It’s one thing to come up with a definition of what constitutes a civilization, but it is another thing to come up with a definition of distinct civilizations, and this happens to align more with the OP’s particular concern. For this I would say that a distinct civilization emerges when the organization of people has a distinct sociocultural superstratum. That is, a particular culture or group of cultures is imbued in the way the civilization organizes itself. For example, we would associate ancient Greek philosophy and religion solely with Ancient Greek civilization, and the way they organized themselves went hand in hand with their cultural values (like the ideal of the polis and serving and fighting for one’s polis). This is something distinct from Egyptian civilization, which had more authoritarian and absolutist values in their culture, and that was reflected in the way they organized themselves (a pharaoh who acted as a god-king). This doesn’t mean that cultural values in a civilization don’t change, in fact they do all the time, especially when it comes to political organization. The Hellenistic era and the rise of Rome deemphasized the role of the polis. The role of the Egyptian king/pharaoh fluctuated over time. However over time things gradually drift apart and cultural value systems become incompatible. The way a civilization becomes distinct then is pretty much the way any sociocultural system becomes distinct, albeit on a wider scale. The most visible way this has occurred in history is with western Eurasian religions, like Western and Eastern Christianity and Islam. I would argue that Christendom and Islamdom were distinct civilizations, because of the core cultural and religious values that were incompatible with each other, but also united different nations under those cultural and religious values. This also means that a civilization could be distinguished chronologically. One could argue that medieval Christendom and classical Greece and Rome were entirely different civilizations, even though most of us would consider them ‘Western.’ Likewise, you could argue that the modern day secular, industrialized, Anglicized West is distinct enough to be a different civilization from medieval and early modern Europe (though I would personably disagree). I think it also depends on self-perception to an extent, too. There are clearly people who think that their society is a different kind of society from other groups of people, and I think we should take that to be pragmatically meaningful.
I think we should also consider that there is a difference between a ‘civilization,’ a ‘nation,’ and a ‘state.’ A civilization would be the framework of organization and management of urbanized populations. A state is just the central political organization that does the managing. Civilizations tend to form states, though it doesn’t have to be completely divorced from other cultural and religious functions, as was the case for most of history. A nation is a body of people who share a common cultural heritage. A nation-state is a state governing a particular nation. A city-state is a state governing a particular city. You get the idea. Now nations and states work within the framework of a civilization, but they aren’t civilizations themselves. If you were a peasant living in the 1300s, it wouldn’t matter whether your lord was English or French, you were still living in medieval Christian civilization.
These differences are relevant because we have to keep in mind whether the civilizations in the Civilization games are really just nations or states. If you think about it most of them are. We never talk of ‘Polish civilization’ or ‘Venetian civilization.’ We always say ‘Polish nation” and ‘Venetian city-state’
The only 2 factions in the game that could unambiguously be considered civilizations are China and India, both of which have always been a distinct civilization from the very beginning, and both of them have very long and convoluted histories that their representation in the Civ series does very little justice. The very idea of a united Indian state is a recent concept. All other civs in the game are either too vague or too specific. Venice isn’t a civilization. Portugal isn’t a civilization. America isn’t its own civilization, it’s an empire (half-joking here). There is no Dutch civilization. Most European civs aren’t their own civilizations. And then there’s cases like Greece where we have a modern nation named Greece and then we have the ancient Mycenaean and Helladic civilizations which were quite distinct from modern Greek culture. But then there’s Byzantium which is technically a continuation of Classical Greek and Roman civilization but also has the Eastern Orthodox Christianity DLC so I guess one could argue that it’s different enough to be considered a civilization. Then there’s Sumer which I would argue wasn’t a completely separate civilization but the first iteration of Mesopotamian civilization, which also included Babylon and Assyria, and they all shared similar cultures. But then there’s the Maya and Aztecs and the Aztecs I mean they were a pretty temporal state and kinda just built off the altepetl tributary system which was already in central Mexico for a while, so I wouldn’t count them as a distinct civilization. The Maya meanwhile is a little ambiguous because they do have some features that separate them from other Mesoamerican societies but at the same time they do share a lot in common so it’s up to debate. It’s a similar deal with Greece and Rome considered individually, Rome itself was a state that absorbed several civilizations like Carthage, the Iberians, the Egyptians, the Canaanites, and the Etruscans. But most famously they subdued the Greeks and they pretty much copied all of their culture, so because of this I would consider both Greece and Rome to be under the umbrella of ‘classical civilization’ though Rome was initially connected to Etruscan/Italian civilization.
All of this should go to show that there aren’t actually very many civilizations in the Civilization series, at least not unless you narrow down the definition of civilization to include any distinct nation group. But I think this is how it should be. The Māori never circumnavigated the entire globe and plopped down to build big cities and farms. The Romans never developed tanks for their legions. The Maya never built spaceships to explore the stars rather than just watching them. America never existed during the Neolithic. The Russians never got to dominate the whole of Europe culturally and religiously. The point of Civilization isn’t to define what a civilization is. The point of Civilization is for you to define what civilization means for you. You could be the Māori that settles and vibes across the entire world. You could be the Romans that demolish entire cities with your mechanized legions. You could be the Maya that make ever greater advances in science and technology. You could be the Americans existing in 3000 BC. You could be the Russians that truly become the Third Rome. In that case all cultures deserve a chance in the spotlight, because they all have the potential to do great things (and also because having unique cultures is cool), and we should all get a chance to learn about them. Maybe that means I was wrong about civilizations. Maybe it really is all subjective, maybe civilization is what we make if it, maybe civilization was the friends we made along the way. Maybe there was a man who wanted to share his vision with the world, a vision of a game where you get to decide and play and experience these ideas. But we do know one thing. That game was called Sid Meier’s Civilization.