Professional Dog Breeder: someone who breeds dogs for a specific function (other than profit). Functions can include things like performance sports, hunting, herding, service, therapy and beauty pageants. Even a person who is deeply dedicated to breeding a dog to be a family companion can be a professional dog breeder, only they are few and far between. There are many functions for a dog apart from "livestock".
Private Dog Breeder: someone who breeds dogs for profit. This person only breeds companion animals and may pretend to also occasionally breed a dog for a function but will not be well-respected or well-known in the community of that particular function.
Why do you breed dogs?
If the answer to this is “because I love dogs” make a negative sign on your little check-list. It doesn’t disqualify a breeder; especially because that is likely the answer you’ll get from every single private breeder out there. You should get a much more detailed answer and it should be more than liking dogs. Lots of people like dogs and don’t want to breed them. Breeding takes a tremendous amount of dedication. If a breeder says something to the effect of “I like breeding-it is interesting” or “I love this dog breed” and nothing else you can pretty much chalk them up to being a private-breeder. I don’t care how much they tell you they’re a hobby breeder. Ok, then their hobby is producing dogs for money. That isn’t the kind of hobby breeder people recommend and look for.
A breeder should be producing dogs for a function and serious breeders don’t produce dogs primarily for the function of “pet”. They just don’t. This is an historical fact and not something based upon my personal opinion. Dogs are produced for a function and if their function is “pet” that means they’re being produced for money (so really their function is “profit”) and the breeder is a private breeder whether small scale (aka hobby) or large scale (aka puppy mill/farm).
Now a professional breeder does produce pet dogs and should absolutely care about the type of companion they produce, but their answer will include their love for that function and their passion for the dogs being particularly suited to that function. A professional breeder will have an in-depth working knowledge of their dogs and temperament, intelligence and the science behind breeding. They are likely to have a network of supporting breeders and a mentor (or are in the position of being a mentor to others).
A professional breeder is preferable by far but a private breeder can still produce absolutely lovely dogs. For some breeds and hybrids you will really only find private breeders so they may be your only option. You shouldn’t abandon a breed purely because you can only find private breeders.