After 20+ years of having only cats, I got my first dog last year. I had many of the same fears you do especially because one of my friends had the exact same horrible experience you've described.
My first criteria was to get an adult dog that is known to be good around cats, one that is indifferent about them. I think indifferent is better than interest, even if friendly. You can never know how a puppy is going to act as an adult so I would not recommend one. The second thing for me was to get a breed that is known for low prey drive....it's not a guarantee, but it's less likely to snap unexpectedly.
Some people will tell you to never leave you dog and cats alone together, that anything can happen. It took me a while to leave Jake with the cats, but I do it. I think it's a matter of knowing your animals.
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Holly, Misty & their friend, Jake, the dog.
Onyx, Callie May, Maggie & Kobi forever in my heart.
Yes, I know I need a new signature pic...someday...