This is an interesting question, which needs some thought.
First, your interpretation of the second sentence is wrong: it doesn't mean that she killed the cat herself: she did not, but she arranged for it to be killed.
The problem is that the pattern have something done has two different meanings.
Usually, it means 1) cause or arrange that somebody else do the thing. So I had my door painted means that I did not paint my door, but I asked (or paid) somebody to paint it.
But it can also mean 2) suffer from somebody doing the thing. Most times you can tell which, because this meaning is only used when the thing is something bad for us.
So, since nobody wants their goods to be stolen, Jenny had her car stolen must mean Jenny had the bad thing happen to her that somebody stole her car. Conversely, I had my door painted can only have this meaning if having my door painted is something I would not want: compare I had my door spattered with paint, which probably does carry meaning
Your second sentence, about the cat, could mean either, depending on the circumstances and how Jenny feels about her cat. Without context, I think most people would take it to be 2) - somebody killed her cat, and she's not happy about it. But if we know that Jenny is a cruel person, or if we know that the cat is very ill and she does not want to suffer any more, then it might mean 1) (though most people would use a less stark word than "kill" in this case).
So I had my cat killed could mean that you arranged it, if you are a cruel person like that; but probably means that somebody killed your cat and you are not happy about it. I don't think many people would say this though, possibly because of the potential ambiguity. I think most people would say somebody killed my cat!.