A barking dog could be legally found to be a nuisance, and your local ordinance as you have transcribed in your question, apparently recognizes this. However, the ordinance distinguishes a barking dog as a "public" nuisance and so your local government/code enforcement authority will have the power to take some action. You need to be proactive in making complaints and documenting the frequency (dates/times/duration) as well as how loud the barking is. You might want to consider recording the incidents in some way to document the dog's barking and provide copies to code enforcement.
The other attorney is correct, that you might want to try to communicating directly with the dog's owner to solve the problem, but you may have already done that as well.
Ultimately, whether the barking dog is found to be a public nuisance is a determination that must be made on a case by case basis. So having kept a record of dog barking frequency, and having made recordings of how loud the dog barks are, could be important pieces of evidence that the code enforcement board would use in making that determination. You should also consider organizing your neighbors to join in filing complaints if it's really a bad situation.
My answers on Avvo.com are intended to help clarify the legal process in your situation to the extent possible on this website. Criminal law can be very complicated and very serious. Every case is different and my answers are not intended to serve as legal advice unless so stated. If you have been charged with a crime you could be placed on probation or face incarceration. Please contact an attorney if you think you need legal advice in a criminal matter.
You've already asked this question - the answer is determined by animal control and the courts. I suggest you speak to a local attorney. Your local and state bar associations can give you a referral.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.
First off, have you spoken to the neighbor about this problem? Sometimes people are not aware of their dog's barking or howling during the day when they are gone. The dog is probably bored and reacting as dogs will. Suggest the neighbor keep the dog inside during the day or have a dog walker come to make sure it gets exercise and can relive some of its stress.
If this doesn't work, you should check to see if your community offers a mediation service for issues like this. Often there is a low cost service for neighbor disputes. If not, then offer mediation to your neighbor and if all else fails, check with a local attorney. There may be other ordinances regarding noise that apply to the dog barking as well.
Good luck. I do hope you can resolve this without resorting to the courts.
This response is not to be construed as establishing an attorney-client relationship, and provides general information on the subject at hand only.