This is a very interesting question, and it turns out the answer is complicated. You are right that Dr. Miklosi’s work suggests that specific barks are associated with particular contexts. He has concluded from this that a diversity and increased prevalence of barks has evolved specifically in dogs to communicate with us. You are also right that this would suggest that we should find some strong genetic signal associated with variation among dogs in the prevalence of barking and also, perhaps, the diversity of their barks. However, my work suggests that barking occurs in times of conflict not only in dogs, but in all other members of the genus Canis. It also occurs in many other mammals and birds. Given that barking occurs in times of conflict throughout the genus Canis, another possibility is that dogs bark more often simply because they are in conflicting situations more often than wolves. This is where we get into the differences in “emotions” that you brought up, in addition to another variable: the environment.
Wolves rarely experience conflict in their natural habitat, for example, the approach of a potential predator at a den site. By contrast, dogs experience conflict frequently. They are often unable to flee from frightening things, because dogs are inside a house, or behind a fence, on a leash, etc. Dogs also are often unable to approach things they would like for the same reason, resulting in just as much conflict, indicated by a higher pitched, more tonal bark. In part, then, dogs’ barks maybe more frequent and more diverse simply because dogs experience frequent and variable reasons to bark.
But this difference in environment is not the only reason for increased barking in dogs. Dogs also have a shorter flight distance than wolves, and this does likely have a genetic component to it. Because dogs let scary things get closer they are more likely to get into situations where the scary thing has gotten too close for them to run away. If they turn away, they could get grabbed from behind. This is most often the situation in which you see free-living dogs bark. Dogs that live in our homes also learn to bark. For example, if you have a house-trained dog and you forget to let it outside, there is a good chance it will bark, because it is conflicted over not being able to go outside when it has the urge to urinate. You, hearing this bark, remember you have neglected to let the dog out and open the door. Your dog quickly learns that this sound (which previously had no particular meaning) causes you to open the door.
While there are genetic components that contribute to these things being possible in dogs and less possible in wolves, these may or may not be why different dogs have differences in their bark prevalence, or type. To make things even more complicated, the size of the dog greatly affect the sound of their bark. So, while I am sure there are genetics involved in differences in dog barks, why a dog barks in a particular situation and how it barks is extremely complex and would thus require a finely tuned questionnaire to get at the trait of interest. We have yet to develop such a questionnaire, but we are always eager to hear participants ideas for questionnaires and we will certainly keep this interesting topic in mind for the future.