If a cat brings a souvenier home, it's an offering to you and is a very high compliment indeed. The fact that it's either dead, barely alive or partially dissected is immaterial, but it's a cat's way of telling you they are returning the favour and that you look hungry. They do the same for their kittens.
Leonard was a very chilled creature, who just used to sit and watch stuff all day long. The only things he'd bother to catch were frogs, who were always physically unharmed by their ordeal, but would often pass out through fear.
First thing I'd know about it would be the strained, pleading "Meeeeeoooooowwwwwww!" which meant "Help! I've got something in my mouth that I don't know what to do with!" so I'd have to go outside to him where he'd present me with a very bewildered amphibian.
Jasmine (Wookie sized cat) was a killer. She'd stalk stuff and bring home the remains frequently, although if you told her she was beautiful, she'd roll on her back and demand you tickle her belly. A real Jekyll and Hyde character...