I've won the Diplomatic way once, by accident. Wanted to see what would happen if I held the vote, and had good ties with most of them, and had killed a few so was the biggest on map... so they voted for me.
All but the second bigest who voted for themself and the one that had been reduced to an island and hated me... for reducing them to one island.
Usually don't hold the vote when it asks now. One time I lost... ICBM's put a big hit against everyone when I thought there was no majority.
Never done a game where that was the only condition to win though. Have also won the space shuttle way, but need to do it again as I never saw any movie or animation when it finished or launched.

Never even thought of cultural... not even sure I know the conditions on that one hehe...
One thing about the Carthage guys is they get the pyramid early. So don't even have to worry about pottery at first really. Put 100% tech on the temple thing for some good BC culture, numidian mercs are next tech. Usually get 2 - 3 warriors/settlers/workers. Then keep producing till you see someone else starting pyramids, or just start them early... expand to get either more land, or specific resources depending on where I am/how many opponents n stuff. Often able to have a good half dozen cities or more when the pyramids hit, then more expansion, defend till I decide to battle.
The early temples give great culture later on... and I always get the pyramids. Usually have the highest culture later on no matter what they build. I try to get Theory of Evolution first, and use that on the longest tech to learn at the time, then go back later for the riflemen and military build. Communism is what I usually end the game with, it's better for war. Often keep Dictatorship till Democracy, sometimes I'll go monotheistic or whatever that is depending on if someone's whacking me hard early.
Wonder if that'd win Cultural too?