That by-law is badly drafted. You don’t “keep” animals on the community property, so usually the by-law will say something about the keeping of animals in a lot, and then additional provisions about animals on the common property, i.e. how they must be controlled etc.
12.1 and 12.3 are inconsistent because 12.1 refers to community property and 12.3 refers to a lot.
Anyway – I am not sure your interpretation of it is right. If you go to the intent of the by-law, looking at it as a whole, it seems to be that an owner can’t keep an animal in their lot without written permission, and that when an animal is on the common property it has to be on a lead. This is because 12.3 refers to an animal being kept on a lot with permission, which presumes that permission must be given.
Perhaps the best option is to ask the strata manager, and at the same time I would point out the deficiencies in the by-law, because as it is it doesn’t make any sense. It should be amended to avoid confusion.