I'm sceptical when it comes to a difference in slender when you compare Europe to the US.
I've caught many really slender carp in France,in fact most of the carp I caught in France where really slender.
Those big fat European Carp are in fact really fake,I'f you would ask me to describe them.
Those fish are usssually from heavily fished ponds,where fishermen throw tons of food in the water each year just to get the carps attention.
The carp can eat all they want each year and need little exercise to get it,so they get really big.
It's like taking an LMB put him in a pond overfeed him for a couple of weeks and then pose with the fish on a photo,really nothing special about it.
When I go out fishing within my hometown I ussually see carp that you describe like tanks in the watersystems,who are big fat and piglike shaped.
But when I take my bicycle and go to some unknown waters the carp are ussually the same as in your pictures.
Carp just need some conditions to grow really big and because carp are really popular here those conditions are created making those carps huge,In your country carps are trashfish and so those conditions are never created which gives them there average look.