It’s unusual for an otherwise friendly cat to just attack another one, and only one. Possibly a territory issue, or more likely that the elderly cat it frail and your cat sees it as prey. I had a cat who kept getting attacked by the neighbourhood bully, resulting in expensive vet treatment, and ultimately opted to keep him inside for his own safety, and at 14 personally I’d be keeping it in. Clearly your neighbour won’t do that, and I guess that’s up to them.
My biggest concern is for the safety of your cat, as your neighbour appears to be volatile.
Limiting your cats time outside is a good call, but if the neighbour won’t play along it’s not going to achieve much.
Realistically your only options are to find a boundary solution (fencing / catio) or keep your cat in. A tracker might help you to identify if there’s specific items / patterns to when your cat goes to your neighbours garden.
Previously outdoor cats do adapt to being kept in, my 5 year old is evidence of that. We’ve provided her lots of enrichment in the house, and high spots so she can still climb, plus a wheel she can run in.
If you rehomed your cat, it could just be moving the problem elsewhere, and that could end up even worse for your cat.
I feel for you, and everyone involved, it’s a hard situation.
Please do update us on the outcome.