Throwing caution to the wind, and speaking to a British exam question in an American voice, I'll suggest that perhaps sense, which I agree would have been a more felicitous choice, invites the reader to a connotative understanding, whereas meaning calls up a more objective decoding of the word.
Sense, after all denotes perception in a variety of modes: eye, ear, nose, tongue and touch, privatim et seriatim - as well as in concatenation. Sense implies the right-brain, intuitive process - just as the 18th/19th C notion of 'sensibility' implied a balanced blend of sentiment and reason. Where others may look to,lexicography, I turn to Pope: sense is aligned with art.
Meaning, on the other hand, seems to me to spring from the left-brain realm of objectivity. Pope has no place in a quest for meaning, which is to be found in the company of Murray, Webster et. als.
As you say, it's the sort of fine distiction that discriminates among responses at the highest level. As a reader of high-stakes, high-level exam papers, I admire the distinction made in this case.