Those are game models, which shouldn't be taken too literal.

Culture represents all culture of the civilization, not just arts, but also lifestyle, architecture, literature etc. Every tile within borders is considered inhabitated in Civilization and culture determines to which nationality the inhabitants feel they belong. In that sense it represents the assimilation of a people into anothers cultures. Look towards the Romans, who employed strategy like this to keep conquered people in line. Or look towards assimilating a foreign culture via trade, eventually it becoming part of your own.

Influence on the hand is completely different. It represents the nations political clout and power and such is a very real tool in determining borders. Think here about the political maneuvers used to expand ones territory. E.g. the marriage of Isabella and Ferdinand, which created Spain out of Castille and Aragon. Hitlers actions before WW2, where he used his influence and power to conquer several territories without a fight.

yes, culture is all you value and not just art, but he is asking about times when culture changed borders. Italy and Germany above are good examples. Music brings our people together.

One could argue that a common culture was a reason to form the Indian, Chinese, Japanese states too. Even if there were good political and military reasons, would a maharajah, a damyo or a chinese warlord accept any central government if the larger entity were not seen as common cultural ground, as a mother nation, even when the previously existing strong unified government was long gone?

In the indian case, among all local enmities and differences, only the muslim desire to a muslim state was strong enough to prevail and generate Pakistan. As religion is the most fundamental part of culture (if culture is what you value, then religion defines your values about the most fundamental aspects of life), it is culture at play.

Switzerland

Why did individuals cantons joined? They did not join all at the same time. Mutual defense is a reason, but the federative, autonomous, republican culture must have been important too. Would they join a feudal kingdom? heck, they even attracted and kept cantons with different languages and religions!

Texas, USA, America

Texas proclaims independence, and after a few years, join the US. It certainly helped that they were American colonists from the beginning, and some of them might already be aiming to USA statehood beforehand, but anyway: If USA somehow were changed culturally or politically during the few years of Texas independence, they could have tried to stay independent.

And if Porto Rico runs a plebiscite again and joins the US, does it count?

Latin America had urges of union with their cultural brothers too (see Bolivar, San Martin, the Union of Central America). But the distances and communication difficulties over the Andes and different seas, a decentralized history under viceroys, etc, all hampered their project.

Colonization as a mechanism

When I played that game I thought on these lines: my people emigrate to these backward neighboring regions and those primitives are so in awe that they don't even resist. But certainly IRL it is not so simple.

You may think of colonization of a neighboring region as a mechanism to implement "cultural annexation". Certainly, a culture that encourages this kind of emigration is helpful or even necessary, besides a neighboring region underpopulated, unable or unwilling to resist. In this case, russian expansion into siberia, overall USA expansion in the West, Brazilian expansion in south america are examples where the native population might be discouraged of resistance by the superior tech, numbers and culture of the invaders. Sometimes this expansion happened even with not much direct help or interest from central governments.

Present day Chinese emigration in the Far East Siberia (which some think will end up in a chinese siberia someday), is a (futuristic) example too.

Many muslims see the current emigration wave as cultural annexation too (they have even a specific word for emigrating to change the demography). The Europeans are considered too naive and decadent to realize what will happen when their countries have a muslim quasi-majority. In this case the difference is not superior tech but simply a culture that still have children and is proud of itself versus a culture with no children and not much will to preserve or even know itself.