I've worked as a zoning plans examiner, and also wrote zoning determination letters deciding things like this. I've held multiple credentials in zoning code function for many years.
As has been pointed out, approach it as you seem to be doing, get the local official making the call to work with you to suggest how to accomplish your goal. The code enforcement person is not usually the highest person having authority, it's the chief building official here in Florida.
Avoid the variance option at all costs. It can cost much money, and there is no guarantee it will go your way. All they need to hear, is that some political city hall gadfly appears before the town council, the city commission, or the whatever, and complains, "you're ruining the community by allowing this homeowner to circumvent the code!"
Getting the zoning determination in writing is what you need to do. If they agree that you can re-name the 'breezeway' to call it an entry foyer, or mudroom, or hallway, and there is some minimal amount of work to be done, as-in being totally-enclosed between the garage and the house, then that's one way to approach it.
If I were you, I would see who sits as an architect on the Zoning Board, the Planning & Zoning Board, the Plans Review Board, and I would pay them for a consult. They know what will 'fly,' and what won't. You may not need them to draft a set of plans for you, but that's one way to go, and you have a reasonable expectation that it will be approved if that design professional does the plans. They will need to disclose their professional involvement if it comes before the panel/board on-which they sit, but the approval should be a formality, from what I've seen.
You could ask the chief building official the same question you've asked of the zoning person, and they should be able to answer things like that, because he/she is probably above the zoning person in the hierarchy of the local government.
We used to charge a fee of several hundred dollars for a zoning determination letter. The benefit to you is that you have an avenue to seek relief if your zoning person denies you a permit, for instance, but you have the zoning determination letter saying you're allowed to do what you want to do. I don't think it's wise to pit two employees of the local government against each other, but if your proposed site improvement meets code, the local government must issue the permit. The fact that your barn (for example) is going to house your collection of motorcycles, while the neighbors on either side of you are 'horse people,' well, tough. Obviously, you don't want to get into a ******* match with either the local government or a neighbor, but you are entitled to a permit if your plan meets the development code. What the code interpretation is can be a matter of opinion, and that's what you are trying to resolve.
Find out what your local jurisdiction uses for online access to their zoning code, and building code, it will actually be the entire code of ordinances for the community, or the county. American Legal Publishing is one, and Municode is another. Then you can research the terms being bandied-about, and perhaps answer your own question.
In the end, I suggest using a co-operative approach with the local zoning person to achieve what your goals are, and it doesn't hurt you to have done your research on the nomenclature and definitions contained in the building code adopted by your local government. Using a design professional well-known to the local zoning and building departments is also worth considering, but try to do your homework first, and then you can ask, "is this feasible?"
I once had a supervisor who was in my opinion, unethical, and I had to tell her on more than one occasion, (tactfully) "no you're wrong." It didn't go-over well with her to hear that, but if someone tells me to sign my name to a permit paper application, and I disagree with the person's order, I will and have told them, "sign it yourself." My ethical reputation is more important to me than pleasing my boss. In rare instances, you may have to terminate employment to protect yourself. That's an entire different thing than what's been discussed here.
Keep working with that zoning person and I suggest getting the chief building official involved as here in Florida they are the 'authority having jurisdiction.'
