Kiba also chases the outdoor cats because they run from him. He's fine with the indoor cats that stand up to him. He's gotten better about listening to a leave it command when he sees the outdoor kitties, but I think he'll always have that instinct to chase a running animal. A calm cat and a running cat strike two different cords with dogs.
So, if your outdoor cat keeps running from the dog...the dog is going to want to keep chasing it. You can work on desensitizing Delilah to the cat as it is running, but unless she has the right temperament she'll likely always have that instinctual, "ZOMG, it runs, must chase!" reaction. And the desensitizing can only happen with constant supervision with no chance for a cat-chasing relapse.
You can't train her not to chase the cat. You can train her to "stay" and things so that while you're watching her, she might not do it, but if you go away, she's going to chase the cat.
Well, that's not quite true...Rose used to be a cat killer. She would chase outdoor cats with the intent to kill and I know of at least one feral cat she did indeed catch before my mom could do anything about it. Rose now lives happily and safely with two indoor cats and three outdoor cats. She doesn't like the cats (the cats adore her and rub up against her all the time and she gets this look of disgust on her face), but she knows to not chase them now even when we aren't supervising. Once she realized cat chasing = very, very unhappy humans she tolerates the cats. That is just the kind of dog she is.
Same goes with Chloe. She chases all unfamiliar cats and even on occasion our cats if they act spooky around her, but she's perfectly fine with them 99.9999999% of the time and she
loves kittens. Chloe's cat chasing is more territorial based - there are "our" cats and "strange" cats and strange cats are to be driven off of property.
It also helped that our outdoor cats are dog friendly if the dogs are friendly - once they realized we wouldn't let Rose (or Chloe) chase them the stop running and become very lovey dovey with the dogs. When they stop running, the dogs loose the desire to chase them and all is well.
An outdoor farm pet? You do realize dogs are social creatures and tend to get destructive and bored when left alone right? So chasing cats is a good way to stem off boredom. I am all for farm dogs, but why get a dog to live outside if its just a pet?
And, keeping outdoor dogs is possible...Blackie and Rose were outdoor dogs the first 10 years of their life. (After Blackie started having medical issues we moved them indoors.) Even today, Rose is more than happy to spend all day outside and come in only to sleep at night. We have ten acres and even thought we aren't surrounded by empty fields anymore, she's old enough now that she just sticks to the yard and all is well.
While I don't like the idea of a single dog being kept outside without any buddies, if you live in a really remote place without any close neighbors, I don't see why dogs can't be happy outside. We're more populated out here now so having that kind of freedom outdoors is a no-no, but when Blackie and Rose were younger it was just us, maybe two neighbors down the way, and fields upon fields with any non gravel main road being miles away. They were very satisfied pooches being able to run, swim, dig, and hunt to their hearts content all day long and we never had any problems with them at all.
I don't know if that is this person's situation, but I'm not going to knock them for keeping their dog outdoors when I don't know what their set up is like.