You can call your dog anything you like mate thats your choice., me I dont call em terriers til dug to, thats my choice.. lad i know keeps a dog, looks likes a terrier, acts like a terrier, chases rats an rabbits...wont go to ground tho. You might call that a terrier, but i dont, I call that the deceased.
Seems strange that you wouldn't want a dog that could do both. At least that way you would know that you had a dog that could track if need be. It's pretty simple to trash break a dog. Is it not common for your dogs to prefer toothed game to chasing bunnies once they've had a taste o