Pretty much a .45 Colt - Long Colt thing, except .32s are less popular now and people haven't got their minds made up.
I looked up those advertisements so I could blow them up, and sure enough, the ammo company labels are ".32 S&W" nowhere saying "short" on the box.
The dealers are saying "short" to try to inform their customers. It has gotten some folks confused, shooting .32 Short Colt in a .32 S&W.
There are a couple of other misnomers, heard less often now than in the past.
The .38 S&W was known as the ".38 Regular" to distinguish it from the .38 Special.
The .32 S&W Long was known as the ".32 Special" because it is proportioned a lot like .38 Special and to differentiate it from .32 S&W and .32 S and L Colt.