I sometimes forget the quite young demographic of this forum, & am surprised by threads like this one.
The term "zinc-carbon" is widely used for "plain old primary cells" of the type which were used in their billions prior to the commercial availability of the plethora of cell chemistries we see today.
I can't really remember them being called "carbon zinc".
https://en.wikipedia.org/wiki/Zinc-carbon_batteryAs Wikipedia point out "Zinc-Carbon" is a misnomer, as the the active electrodes are the zinc case, & a quantity of crushed/granulated magnesium dioxide, which, classically was contained in a cloth bag attached to the zinc rod.
The normal,"D","C","AA" etc cells, (as well as the various batteries using this chemistry) do have a somewhat shorter life than alkaline cells, but long life "zinc carbon" cells have existed, (& maybe still exist), such as the "No 6" cell, used for many years in manual phone systems, & for many years after that wherever long life cells were required for relatively light duty.
When I say "for many years", I mean it in both senses.
It was not uncommon to find No 6 cells which had been in use for decades.
They are huge---about the same size as a 750ml wine bottle,minus the neck.
It was this, & some variations in chemistry that allowed such longevity.