Well, this is just what I don't want to do - culture would still define borders, just a new way of generating it would be introduced. This doesn't work for modern times - no monument or library can match the strong infantry and tanks when it comes to controlling land and resources. Also, the borders wouldn't be fixed during peacetime, since normal culture would still flip tiles.
True, No monument can match troops But your troops are sometimes outmatched by the strong will of the inhabitants answering with fierce resistance!
when you understand culture = nationality (in a city tile it's even called so) it's quite consistent with modern times. with plebiscites like after the world war II in europe some borders were changed according exactly this: culture. ok, the borders in the east were determinated by other aspects (but still since several cities changed their owners too it's not inconsistent either).
anyway, you cannot deny that nationality plays a significant role in determinating todays borders.
another aspect of nationality/culture is that you can't use enemy roads. i always understood this as the inhabitants of these tiles being uncooperative, granting you no support but resistance instead thus slowing your troops down. so it would be most convenient if you need much longer to claim territory with huge culture than territory with very little of it. in addition to this i would even go so far that a very high culture on a tile should even hurt enemy troops passing through it. but that's just me i guess.
the only thing about culture that is a bit strange is how strongly it spreads in territory beyond your borders, in that i agree. especially if you have no trade or any other agreement with your neighbors. in that case it should not spread in your opponents territory at all! with open borders agreement and good diplomatic relations culture spread on foreign territory could come up to 50% of normal culture rate. also foreign culture should
never spread on tiles with a foreign fort so they can be used to fix ownership. so change that about culture and you'll have your fixed borders.
when you now imagine you could diminish enemy culture on tiles with military this will effectively result in claiming the land by repressing foreign populations resistance. furthermore you would have forts to claim territory beyond your cities domain.
i think you should consider to adapt cultural (nationality) borders behavior to the idea of borders you have instead of making a completely new mechanics for them.
Instead of paying gold for claimed land - each turn you'd automatically lose any tile that has neither your culture nor your units on it. When you lose it, the player with the highest culture in the plot would get it or, if no culture existed, it would become unowned. This will make tiles farther from cities more difficult to maintain, especially in early times (without airlifts and fast supply roads). Also, the infamous "razing of barb cities" problem will be resolved.
or this. but beware if you want to diminish culture on tiles you will be forced to go throgh every tile of the map and check for every nationality each turn. this can mean quite a lot of performance on large maps with many civs i guess.