I own a rat terrier, know lots of other rattie owners and also know folks who own toy fox terriers and smooth fox terrier. Here's why you're likely confused:
1. Many dog websites don't list the rat terrier at all. That's because they use what the AKC sets as "breeds." But the AKC recently allowed ILP or FSS registration of rat terriers. So the websites that don't list the rat terrier are either out of date or incomplete.
2. The Smooth fox terrier looks nothing like a rat terrier. Muzzle, tuck-in, coat--all noticeably different.
3. The Toy Fox Terrier (there is no "Fox terrier" according to AKC) is tiny. Some rat terrier breeders insist that a TFT is just a miniature rat terrier. But let's put it this way: rat terriers come in 4 sizes...toy, miniature, standard and decker. My rat terrier is a big miniature (not quite big enough to be a standard sized rattie). He weighs 17 pounds and the maximum size for a TFT is about 6-7 pounds. A decker line rat terrier gets up to 40 pounds! So clearly rat terriers aren't toy fox terriers.
4. Rat terriers developed in the US. American farmers wanted a dog that stayed close to home, was great for dealing with rabbits and other vermin (mice) or critters that dug holes that cows and horses might step in (badgers, muskrats, etc.) so they took the Manchester Terrier and the now extinct Black and White terrier and mixed in sight hound (to react to small movements) like Greyhound and Whippet, and scent hound (to track well) like Beagle. Some farmers wanted dogs that went to ground well and bred in some dachshund. Some wanted to do modest herding of goats or chickens so bred in Corgi. About 110 years ago, nearly every farm in America had a rat terrier. They were a small-medium sized dog that required little upkeep, stayed home, was hardy, great at its job (quick, agile, great sight and scent, good mousing instinct).
The rat terrier got its name when Teddy Roosevelt became President. He had a mongrel terrier named Chip with him when we took over after McKinley died. Roosevelt discovered that Jefferson's old abandoned greenhouse was infested with rats. Teddy set Chip loose. 8 hours later, Chip had killed over 1,000 rats. Roosevelt began referring to him as "my rat terrier" and he bred Chip. Chip died during his second term and is now buried on the White House lawn. But Teddy Roosevelt is regarded as the patron saint of the breed.
In Australia, a relatively similar dog appears to have developed under somewhat similar needs: ranchers needed a dog that was good indoors and out, agile, hardy, not too big (didn't require a lot of food and easy to lift and move around), short-coat, great vision and scent. So they too took terriers and mixed in sight and scent hound and ended up with a dog that looks very close to a rat terrier (but doesn't have the same genetic diversity or range of coats. Thus the Mini Foxie.