They can't keep their cat inside? I'm going back and forth between thinking your neighbors are lazy about taking care of their own pets and don't really care about their health.
Very few people in the Netherlands keep their cats inside the house. No-one keeps them locked up outside by my knowledge. Many people who have cats have a cat pop door in their back door and tge cats can go out whenever they please (other neighbours).
Nowadays there are many people who keep their cats inside at night (like the neighbours with the hungry cat). This is a progress.
All my neighbours care a lot about their cats. Whenever one is sick or hurt they go to vet. And they go to the vet every year for the recommended vaccinations.
So according to your neighbors, the burden is on you to sit outside your coop with a non-lethal weapon waiting to dispatch their pet instead of them taking responsibility for its disruptive behavior. Do they think you have nothing else to do with your time?

sort off , but only if I want to.
My chickens come into my neighbours garden too sometimes. They don’t mind. So who am I to say they have to keep their cats inside.
In general people here tolerate other animals like cats on their properties as long as they do no real damage. Living in a country/town with do many people living side by side, means you have to be tolerant to habits and noises of other people.
Dogs are another story. Dogs are not allowed to free range outside their own garden. So everybody walks their dogs on a leash. And there is an obligation to pick up the poop if the dog poops on the pavement or grass where children might play.
In my town there are 5 off-leash area’s where dogs can play and poop freely within a fence. People with dogs pay extra taxes to make it work. Forests and natural parks often have restrictions for dogs too, to protect wildlife.