The Jack Russell is really, REALLY fragmented right now. It used to be that there was just a "Jack," but now there's (and this is just registered breeds) the Parson Russell, the Jack Russell, and the Russell. And within those are the shortie Jacks, the Irish Jacks, some people have a go-to-ground strain and others have more of a fox-terrier strain... it's ridiculous, honestly, and everybody kind of realizes that but it's a bit out of control with everybody thinking they have the One And Only breed that Reverend Russell himself would have wanted to own.
I have a JRT cross, mom was a shortie Jack and dad was a dachshund, who came up from rural Tennessee as a rescue. And I mean REALLY rural. He was born in a barn and I brought him and two of his siblings up here to foster when they were eight weeks old. He's the type of dog that I used to know a bunch of when I was a kid. They were always in the horse barns and they were quick to bite, quick to hunt, cussed awful dogs who would come trotting out of a stall, bite you on the leg, casually trot off again. He's got the ratting instinct (and therefore chickening instinct) turned up to eleven, ALWAYS. He's here because he was completely unadoptable because he bites everybody (not biting down - snapping). He'll live out his life here where I can keep an eye on him and because my kids are dog-savvy and don't start running around screaming if he snaps near them. (And just to be clear - yes, we've worked with him for years. He's always going to have an extremely low bite threshold; it's not a training issue.) He is a fantastic small-predator dog but cannot ever be allowed to get inside the chicken fence or coop. He spends hours a day standing at the fence vibrating with excitement every time they move. We have our dog fence built completely around the bird's fence - the birds are like the hole in a donut - and this allows him to pick off the rats or possums without killing the chickens.
My good friend, who also has Cardigans, also has a little terrier as her extra dog. She has a registered Russell Terrier, and he is the most social, lovely, sweet dog in the world. He can rat, he will rat, but he doesn't HAVE TO rat the way my Bramble does. Russells are sort of medium-short; they're supposed to be longer than tall but they're not dwarfed the way the shortie Jacks are.
I also know the AKC Parson Russells, who have been show dogs for a while now, and they have considerably more family-friendly temperaments. They're the longer-legged fox-terrier type and are not as natural as ratters.