Apparently European Rabbit was introduced to Great Britain by the Romans. I would still consider them introduced but I’m guessing the ecosystems have pretty much balanced out by now.
A similar question I have is whether something like Cattle Egret can be considered native in North America and other places they’ve spread. They crossed from Africa to South America, naturally as far as anyone knows, and then spread to North America within the past century.
As for wild vs cultivated, that has been discussed quite a bit here. I think it’s a lot more complicated for animals than plants but generally I think they need to breed after being introduced, and then the following generation can be considered “wild” (although there would need to be multiple generations to demonstrate an established population).
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