Correct, they aren't the same, but usually only larger airplanes (747-400, A380 etc.) have a first class and only for very long haul (> 12 hour?) flights.
There are quite a few non-US airlines that have a first class on much shorter flights, like Europe to the East coast or Europe to Gulf countries. Sometimes in single decker planes.
I know people who fly first class between say, Heathrow and JFK, mostly for the VIP hand-holding at both airports, that can save you a lot of time.
They also used to have first class on a daily 747 between London and Moscow, around 4 hours. (I'm not sure if they still do, but they did in 2014 and 2015 because I flew it both years!). Not to mention the cash cow East Coast USA destinations or upmarket bucket and spade Caribbean flights, none of which are 12hrs.
See e.g. https://onemileatatime.com/emirates-a380-first-class-vs-busi...