Which terrier is most useful against rats?

I'm thinking of small, sturdy terrier breed that will hunt rats and mice in my suburban garden. I'm thinking of Fox Terriers, Jack Russel terriers or Jagdterriers. Of these three, the latter are most often used for hunting around here, it seems. Fox Terrier seems really popular as a companion dog. Jagdterriers appear most "serious" about the job, but also difficult otherwise; they're either advertised by breeders who are also hunters, or offered for free, haha. Jack Russel terriers seem to be only kept as companion dogs nowadays, and more often in cities, than in houses with gardens, or in the countryside. Seems like Fox Terrier is most suitable of these three, but I wonder if you have any other experience or ideas perhaps.
I think Dachshunds were also kept for this purpose but nowadays I think they're just companion dogs which enjoy digging holes, not much fun for a gardener...

The Rat Terrier

The Rat Terrier is an American dog breed with a background as a farm dog and hunting companion.They share much ancestry with the small hunting dogs known as feists. Common throughout family farms in the 1920s and 1930s, they are now recognized by the United and American Kennel Clubs and are considered a rare breed.Today's Rat Terrier is an intelligent and active small dog that is kept both for pest control and as a family pet.

The earliest known record of a rat-catching dog is that of "Hatch", whose remains were recovered from the Mary Rose, the flagship of Henry VIII, sunk in 1545 and re-raised in 1982. Hatch is thought to have been a mongrel, and was brought on board to control the rat population.

https://en.wikipedia.org/wiki/Rat_Terrier

A neighbor had a herd of Rat Terriers. While they lived next door to us, we had no mice or moles.  I do need to add that we enjoyed the sound of them yipping as they ran through our yard early in the morning.  Others may have different feelings.  I also need to mention that very much like herd dogs they seemed to work better as a group. I am sure one would be effective .... three would be deadly.  Also, as a group they could deal with much larger dogs when, solo, one would not stand a chance.

Of course, rat terriers! But they're not popular here :(
I guess Fox Terriers (especially the smooth variety) are taking their place. There are also Fox Terrier mixes, which I would even prefer, if reasonable (matching personalities / purpose, and conformation), because of health.

There is also Manchester Terrier - I know a few good breeders but they are focused on sports (dog trekking, agility) so I'm not sure about their instincts.

I had a Manchester Terrier who was a great mouser.  Really quick and deadly.  She appeared to hunt in sight, though, and as her eyes clouded with age she no longer hunted.

Oddly, I knew a Newfoundland that was a great rat catcher also.  

If you were in the US I would recommend Rat Terriers or their short legged cousin the Teddy Roosevelt Terriers but neither are very common outside the US.  
You are right Jagdterriers are probably a bit much for a garden ratter.  Honestly almost any small terrier picked carefully could work same with dachshunds.  Let the breeders know what you are looking for ie prey drive and desire to hunt/rat.  Terriers aren't the only breeds with ratting instinct.  Some Toy breeds do, as do many Schipperke and some Keeshond/wolf spitz.  What ever you pick do make sure you can live with the dog.   Tolerate their grooming requirements and shedding and for most of them their stubborn independence.

Sure, I always had dogs but I never thought of them as part of the permaculture ecosystem, like other animals and plants. Although of course I chose them to suit my lifestyle.