13 week old puppies don't pee "on purpose" to make you clean it up. They simply have absolutely no concept that peeing should be done outdoors. Pottying outdoors is absolutely not a natural behaviour for a dog (complete opposite of a cat) is it something that has to be taught. Same for peeing on newspaper - there's no inbuilt instinct for a dog that it should pee on newspaper, it's an entirely taught behaviour - taught by you.
Moreover, even with effective teaching, your puppy is going to be six months or more before she is capable of understanding that she must potty outdoors only. Prior to that, she simply isn't capable of fully understanding. So you need to give up on the concept that she's somehow doing it to make your life harder. It just doesn't work that way with babies. LOL - I can understand the frustration though. But you just have to try to be patient and work on getting her to understand that pottying outside is a Good Thing.
Personally, I would give up on the newspaper training thing. That just teaches the puppy that peeing indoors is OK, and (as you're discovering) getting a baby pup to make a distinction between newspaper and carpet really isn't that easy. It is better to simply take her outdoors every single time. Start as you mean her to go on, in other words. That is how she will learn the quickest.
What's your long term plan? That the dog will use the patio to toilet on? If so, then you need to take her out there for every single potty stop. And praise her madly when she potties there. If you don't want her using the patio, then you should be taking her out to wherever she is to be allowed to go instead. It really is the fastest way. Babies aren't that clever - and human desires for pottying to take place in particular places isn't an easy concept to grasp at the best of times. Best that you make it as straightforward with as few opportunities for confusion as you possibly can.