Authority having Jurisdiction

"Authority having jurisdiction" does not refer to a single, specific person/role so I think that's where you are getting tripped up. In general, it refers to any party/group that has the final say/authority over a specific set of rules or compliance. It also reflects the fact that only that party/group has that particular ability and can stop work/issue penalties for non-compliance. For example, in the US a local Historic Preservation Board might have authority over modifications to historic structures, local EPA Departments might have authority over remediation to brownfield sites, etc. So in the example question above, the "best" correct answer would be B. I get how you could argue that some of the other answers could be correct (only really Owner could draft C; CxA is primarily responsible for A, etc.). However, all of the other answers could actually be completed/reviewed by a variety of folks. Additionally, those items do not have any official stamp/sign-off by an outside party to move the project forward (like code review/permit approval does). So my advice is to always consider the "Authority having Jurisdiction" to refer back to code/zoning/approval processes/groups. The particular group in question can vary based on which specific item you are looking at (as different groups have different jurisdiction).

Kristina is correct and has a great summary of the AHJ. In the US and the construction industry, the AHJ is always the code enforcement entity. The reason we like to use this shortened form is there are many different individuals who have this kind of power over a project. Fire Marshalls, Elevator Inspectors, Building Inspectors, Plumbing Inspectors, and more. Any of them could hold up a project at any stage as related to code conformance, right up to the occupancy permit that allows the owner to use the building.