I think, Milly, that you need to wait and see.
You acted irresponsibly and unfairly in the first place by expecting them to contribute to containing an animal you bought. It's not a matter of whose fence it is. This is specialist animal containment and the animal owner's responsibility entirely - I'd bet at least 80% of boundaries in this country wouldn't contain a largeish dog.
So whose it is is not really the point.
Containing it IS the point.
So, we;ve dealt with what you did. And yes, the dog went, and problem solved.
Your neighbours however also need to be aware of their previous behaviour in terms of the fact they threatened you when your dog got into theirs, and now if their dog gets into yours, they can expect to be on the receiving end of the same.
My feeling is that they will take measures to ensure this does not happen. But if they don't, i suppose your only and best option is to share the cost of a fence.
If their dog actually causes damage to your property of course you may have a case against them, to make them pay for containment for it. But until that happens just see what they do.