The short answer to the question in the title of your post is "Yes".
The more long-winded answer is that you're liable for compensating the owner of 'property' (which includes cats) for any 'damage' (including 'injury') to that 'property' caused by your dog through 'negligence' on your part. i.e. you couldn't be successfully sued if your dog caused damage or injury after a burglar had let it out because you hadn't been 'negligent'. However a court would almost certainly rule that, through failing to tighten your dog's collar sufficiently, it was your 'negligence' which ultimately resulted in 'damage' (= 'injury') to the cat owner's 'property' (= 'the cat').
If you want to 'tough it out' you can see what happens if you simply refuse to pay. In order to force you to do so, the cat owner would first have to send you a formal written demand for payment, stating that court action would follow if you didn't meet the demand. Then, when you didn't pay, they'd need to use the small claims procedure to get a court order against you. (If you still didn't pay then have to go back to court to seek an enforcement order, possibly involving the use of bailiffs).
So you might get away with not paying simply by saying "Tough luck, mate" and hoping that the cat owner won't be prepared to go to court. However you'd be risking ending up having to pay more (through court fees) and possibly damaging your credit score (through having a court order for non-payment of a debt against you). It probably also wouldn't do much to keep the peace within your family.