Well… no. I thought my suggestion was pretty foolproof 
I would just go direct - no pussyfooting (ha unintended pun) around. Life’s too short, just ask her outright and say you weren’t happy about it but…blah blah.
I do see exactly what you’re saying.
The only other suggestion I can offer is to perhaps visit the local pet store, or Amazon, to find a cat scratching post. You don’t need a giant elaborate cat tree, a post should suffice. Rub some catnip onto it and leave it in a fairly prominent corner of your rental, where neither the cat, nor the owner, can miss it. Might help save your area rug.
This way, you’ve addressed the problem without “addressing” the problem.
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The best reason to keep aquanet in stock is it removes ink, at least from non porous surfaces. Pen explodes on coffee table, spray into paper towel and wipe repeat until clean. When my four year old (now 10) on the way to Disneyland decided to write on my tan leather seats as opposed to canceling her birthday out of anger I stopped at CVS and picked up a can of Aquanet.
RR
I wet the paper towel and rub it off, if its a lot then more paper towels more spray. I wet it up pretty good. That is an exact measurement…
RR
Edited to add: it may take the shine off of a wood table. It’s a trade off you can oil or wax the table after
Yes–hairspray works wonders for ink. Aquanet will always hold a special place in my heart (80s dos were tough!), and I’m glad to hear it is still around.
Oh, ETA–
Here is a link to the door scratcher we use–and it comes with catnip that is highly recommended three felines I know quite well. There are a million different styles of these. This is just the door one we use. Maybe this will help @cabinhost as @Chloe suggested.
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Oh, wow! There still is a use for Aquanet. Who knew. I have to wonder what’s in that stuff! Thanks for the tip. 
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Every cat owner is different. Some people live in wrecked homes and don’t think anything of doing damage to another house because it’s what they’re used to, other people are fanatically clean and organized no matter where they go - we can’t predict how the human owner will behave just because we are also cat owners. If it’s important to you that they not let the cat scratch up the place, and you think a cat they brought did the damage, tell them you noticed the damage, ask if it was from a pet (no need to guess the species), and ask if they will be able to stop the pet from doing said damage when they rent again. You can tell them what you told us, that you don’t mind if she brings a pet as long as she can be responsible for not letting it damage the property and making you whole if it does.
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