I've always seen a civ victory as being the equivalent of somehow causing governance to be impossible for all other great powers.
Domination - A country basically being strong enough to take and hold everyone's capital
and actually doing it, self-explanatory. Not the US, obviously - they got as far as Baghdad, the capital of a half-rate Third World country and quit. France probably came the closest to this in the 1800s.
Cultural - I've always understood tourism to mean emigration, actually - a country that has such a powerful mystique that every other country can't keep their citizens from flocking there en masse. Those who can't get green visas spend all day thinking about how to get one, and it becomes impossible to govern them. The US is probably "Popular" with most countries and "Influential"/"dominant" over much of the third world - people would emigrate there given the chance. Britain, France, etc. are culture runaways and aren't likely to be caught anytime soon though.
Science - people would be scrambling to get on that ship. Politicians from other countries would quit their job and try to pull strings to get on. The rest of the world breaks down. Again, the US - but we've only just teched to rocketry and lack the
, and the will, to even build the Apollo Program.
Diplomacy - where the entire world wants all great powers acknowledge the dominance of one country by popular demand. Refusal is met by boycotts, embargoes etc, with the effect that all great powers have to go along with the leader. No country has achieved this, or come close - the closest would probably be the US, for one brief period in the 1990s when the Washington Consensus ruled.
$0.02