Yes, your assumtion is correct.
Several things are influenced by the decion to go tall or wide.
(1) Social policies:
Culture requirement for new social policies increase with each city founded (puppets do NOT increase this!). Overall, wide empires will require more culture and receive SPs a bit slower than tall ones.
In addition, some SPs are tailored for tall and wide strategies:
Tradition - tall: Some benefits are only applied for the first 4 cities.
Liberty - wide: Reduces culture cost from number of cities and grants happiness per city.
Order - wide: HUGE benefits for a large empire.
(2) National wonders:
National wonders require a certain building in every city of your civilization, so tall empires are usually faster to build them. In addition, the required amount of hammers increases with the number of cities.
In general, a national wonder is worth 2-3 mediocre developed cities, so tall empires can use the quicker construction to compete with wide ones.
(3) Victory conditions:
Tall empires have an easier time for culture victories, while wide empires are more suited for conquest (at least imho). In general, thanks to the additional happiness in G&K, wide empires will get an advantage over tall ones in the (very) late game - but most tall empires will have finished utopia or the space ship before that really becomes to unbalanced.
(4) Playstyle and preference:
Some people like to play with many smaller cities, others like a few (or even one) mega city. It is just a personal preference, and unlike in previous CIV games, you are not necessarily having a hard time for staying small.
Personally, I usually play OCC or tall with up to 4 cities. Easy to manage for a peaceful player.
As for the different civilizations, everyone can go tall or wide. Some UA or buildings are more suited for a certain playstile, but overall there is probably no CIV where you would have to say "ALWAYS go wide / tall".
My personal list (subjective!) of possible CIV strategies:
Maya - wide (Pyramids grant immense early game benefits)
Ethiopia - tall (their UA makes military defence much easier for a small empires)
Arabia - wide (trade routes & bazar bonus)
Egypt - tall (wonder spamming super city/cities)
Korea - tall (mass specialists)
Carthage - wide (early game trade routes & bonus hammers with free harbours, watermaps only)
France - wide (early game culture bonus can negate the increased culture cost somewhat)
India is a somewhat strange case:
On the one hand you receive 6

per city (instead of 4), but pop

is halved.
So while India starts very slow and has to play tall in the early game, it has enormous potential for going wide once you have the first happiness buildings.
Best tall CIV imho: Korea
Best wide CIV imho: Maya