Can't tell if predatory or just curious

[muttmom]

First: This concerns a dog and a cat.
Second: I don't need reminders about how a dog can kill a cat. I know. I am scared to death about this. I am so, so, so nervous, and believe me, I don't need anymore paranoia instilled in me, or I think I'll just explode. I am almost unbearably nervous about all this.

With that out of the way, here's my situation:
-8 month old puppy of unknown/ambiguous breed origin, turned out to be a pitbull/bull terrier mix upon DNA testing (I know, I know... prey drive, etc. I didn't know what he was when I adopted him). He has been raised with a cat since he was 3 months old, who he respects, who stands her ground and swats him if he oversteps. When she hisses, he gives her a wide berth and moves along.
-My sister moved in with a more energetic and playful cat, who is constantly trying to engage my dog and darting around in front of him. She does not swat or hiss at him, she just runs, crouches and stares at him, puts herself in his path, etc. He is VERY different with this cat versus with my cat.

My concern:
-I'm afraid if he chases after her and/or catches her, he will turn against my other cat, too. I'm afraid he could hurt her, or kill her.
-I can't tell if I'm overreacting to his cues and making the problem worse, if there's a serious problem at all, if he's going to kill the cats and I need to immediately move one or the other out of the house, or what.
-I have no idea what's normal and okay in this context.

His behavior:
-When my sister's cat is around, he stares at her. He usually doesn't move towards her, his tail is kept low/neutral, but his ears are perked, head is raised, and his eyes are fixed on her. I can get him to stop and look at me if I give the command, but then he goes right back to staring. Once she's out of sight, he relaxes again and moves on to something else, instead of pursuing her.
-Sometimes, when he thinks I'm not looking, he gets low and slow and 'stalks' towards her. He stops immediately when I tell him "leave it."
-When he was about 6 months old, he managed to chase after her once, but it was in that bouncy, exaggerated way that dogs do when playing- not in a direct, rocket-like launch. I have never let him have a chance to chase her since.
-He has lightly snapped at the air nearby her, the kind of snap he does when trying to get other dogs to play with him. I immediately corrected him. The cat ran, of course, just to make matters worse.
-I can't tell if this is just typical puppy curiosity and playfulness that he will outgrow with consistent training, or if he's dying to get a chance to kill her.

What I've been doing about it:
-He is in the crate whenever I can't watch him.
-I have been teaching him "leave it" extremely consistently since the day I got him. He's good at it.
-I give him a treat and praise when the cat walks by and he looks up at me instead of focusing on her.... but it's been months of doing this, and he still stares at her, only breaking focus when I tell him to.

I'm having such a hard time finding any information on what normal puppy-to-cat behavior is. Every article I look up acts like anything but the puppy ignoring the cat means the cat is going to die... but I feel like I know for a fact that everyone I've spoken to describes their cat as initially being "harassed" by their puppy until they grew up more. What does harassment mean in this context?

And I am doing everything any article I can find tells me to do- yet, he keeps going back to staring at the cat. Is the staring a big deal if it isn't accompanied by him going towards the cat, raised tail, whining, etc? Am I overconcerned, or not concerned enough? What else can I be doing?
Everything that I can find about dogs staring at cats describes a raised tail, followed by lunging- but he doesn't. So is it innocent curiosity in that case?

[Nettle]

You are correct to be concerned. Right now, the dog is curious and wants to provoke a reaction, but this could easily EASILY escalate into a tragedy. Not the dog's fault, not anyone's fault - just an unreasonable juxtaposition of stimuli. Apart from avoiding a death with your sister's cat, yes you are right that it could cascade over to your cat too, and of course could cause a permanent falling-out between you and your sister.

Training is never 100%. Management is.

You need to figure a way to keep dog and cats separate. How you do it is up to you - you know the layout of your accommodation - we don't. But having the dog in a crate while the cats stroll by is tantalising to the dog and could well, through causing frustration, escalate a shaky situation. So you do need to keep them in separate areas.

Let us know your dog's typical day, what his favourite activities are, how, what and when he is fed and what exercise he gets. We may be able to help here - but the bottom line is that the cats are at risk. Kudos to you for realising it.

[muttmom]

My cat is older, and sleeps 90% of the day, so she isn't strolling by the crate that often. When she does, he doesn't care. As I said, he ignores my cat pretty much. And I put my sister's cat in her room while she's at work during the day (which I'm pretty sure she resents me doing- she seems to think this is all nothing to really be concerned about and I'm just crazy). I work from home, so I'm able to keep an eye on things... mostly. I do need to work sometimes instead of just staring at my dog all day, so he goes to daycare a few times a week and is crated when I really need to concentrate and can't referee him and the cat. He gets more activity in the warm months, walking briskly two or so miles a day on hilly terrain- but unfortunately I learned this year that I have cold-air triggered asthma, and after several really bad asthma attacks, I realized I just can't walk him every day in the winter anymore. So I try to do daycare as often as my budget will allow right now, to compensate.

I feed him Taste of the Wild twice a day in a bit of an unconventional way: I make him stay in the kitchen while I hide little bits of the food throughout the house, and then tell him to go find it. It occupies him for a while and works out his brain.

His favorite activity is probably chewing anything edible (bully sticks, knuckle bones, hooves, etc)- he ignores non-edible chews. His stomach can get quite upset by overdoing it, though, so I have to limit it more than he'd like. He also loves trick training, which I do a lot of with him.

Honest question, though: When is a cat NOT at risk from a puppy? I don't think I've heard of a puppy in my entire life that won't try to bother cats in some way or another. Obviously I'm concerned about the situation, but I'm also living in 24/7 tension right now and actually having nightmares about this, so if this is at all normal, that would be nice to know, too...