Is the debate here really about "what does a 'Civilization' represent?" and less about the specific cases? What does continuity mean? When does "a people" stop being a people?
This is not an easy topic, and probably everyone has their own mesures here. Here is how I see this problem in the context of the game.
Let's take my home country example here. Poland.
It is common agreement that Poland as a state was born in 966 after adopting western Christianity by Mieszko I. So we have a certain starting point. But does it mean Poland was raised from nothing? No. There were Slavic pagan tribes who lived here and are our ancestors. Some people may even consider them as Poles and incorporate their culture, into a larger polish cultural and historical sphere.
How about implications for a game and a possible Civilization pick. For majority of people, those tribes are not part of polish history, so if only we had a fine leader, a tribe with a good story, etc, and everything that is needed to consider some culture as a potentially playable Civ (more on that further) there would be no problem to incorporate this tribe in a game. But for those who consider pre-Christian Slavic tribes that were living along the Vistula River, as Poles, they are already represented in a game by polish Civ. So no room for them here. This example is of course exaggerated, but Poland is a rather easy example of Modern Nation Civ with the exact starting point of its history.
But let's take Italy. Things are starting to be more complicated here.
The modern Italian State was created in 1861 after unification. And again does it mean Italy was raised from nothing? No. But where is this starting point? And here is a problem. The rise of Rome? Some people would agree. Even more, official State Italian interpretation would say Rome was the beginning of what we now know as Italy. Of course, we can say it is overinterpretation. So if it is not a Fall of Rome so maybe a raise of Italian City-States? Perhaps yes. Every Medieval and Renaissance Era Italian City State contributed to modern Italy. But can we say those often very different City-States created something we would call "Civ" or more precise distinct culture? With all differences between Venice and Papal State? Or Tuscany and Sicily? Or was it as we call it here a "blob"? So perhaps the starting point of Italy was a reunification in 1861.
What it has to do with a game itself?
Well from extended Italian definition Italy is a Civilization that started in the ancient era from Rome through Renessaince City-States ending on the modern Italian Republic. From this point of view, there is no space even for Rome in a game. We should have just Italy.
Others would say Italy as a Civilization started after the fall of Rome, so Venice, Tuscany, Papal State, Sicilly are all just parts of Italian Civ. But Rome is a distinct Civ
If we stick to exact starting point of modern Italy then Italy as a Civ should refer only to the last 160 years.
In the end, we would spend the rest of our lives on endless historical or ideological discussions without a good and achievable conclusion.
So what is important for Civilization in a game. In my opinion, it is:
1. Distinct culture we may distinguish from the others.
2. Organized and permanently existing by a certain period of time political structure, or at least strong leader with a good story behind.
3. Historical sources that can describe this culture sufficiently enough to give this Civ UI, UB, UU etc.
4. A good story and theme no other can tell us.
5. Known language their leader can speak and known settlements the game can use.
6. The main theme that can be incorporated into existing game mechanics we want to use.
7. The fact we will not feel a significant loss if we incorporate this Civ and water it down in the larger one.
8. Important to the world history or at last worth knowing.
I used here examples of Poland and Italy, but the list above is universal for every Civ in the context of the game. I hope it helped to explain my point of view.
Why I think Venice is a better choice than Italy
Why I defended "Celtic" Civ and was against considering Scotland as a Celt alternate
Why I don't like the idea of modern Norway and Denmark as an alternate for Viking culture
etc.