Here's the deal.
There was at one time a working terrier, which was used for going after all sorts of critters that was mostly white and came in smooth coat and wire coat.
This group gave rise to the Fox Terrier...both wire hair and smooth hair. The Kennel Club got ahold of it and you basically got two divisions...papered 'show' terriers and unpapered working terriers.
These working terriers came in different types, but often different types in the same litter. And people had no qualm about mixing different variants trying to get good dogs.
Later on people started to slap breed labels on these dogs. You could actually have a litter of pups, and Charlie takes 1 pup while Bob takes another. Charlie ends up registering his pup as a JRT while Bob registers his as a rat terrier. What does that make the other dogs in the same litter?
Deciding to call your terrier mix a JRT X whatever vs a Rat Terrier X whatever is arbitrary. You can choose whichever of those you like.