The cat I had in Italy never killed a bird. Most of the year, she lived inside, and was fine with that. But in summer I'd take her up to the Lago Maggiore for a few weeks, and she could go outside there. She didn't kill any birds, or show any interest in doing so, but she did sort of hypnotize little lizards. They were very fast, and could dart away if I distracted her, but if she was staring at them, they wouldn't move.
And if I took a walk, she'd come along, like a dog.