The new law as I have read it states that "Hunting with dogs" will become ilegal. This means that if I take my dog for a walk and he chases a rabbit I am a criminal. (bill/sables)

I think if you substitute "cat" for "rabbit" in that sentence the distinction becomes fairly straightforward. Taking your dog for a walk, in the course of which it pursues a moggy is a very different thing from going out on a cat hunt

Actually of course in many towns your dog might be as likely to come across a fox, since the more sensible foxes have moved into towns. Maybe they do understand the difference between having to leg it from a mutt on is way home from a trip to the pub and being pursued by a mobb of dogs and humans bent on a kill.