Fish, I think that the phrasing of your quiz...

If you heard that Person A was Person B's namesake, would you assume that:
would direct one to...

(b) Person B was named after Person A
inasmuch as A comes before B.

But to answer as to how I understand the word "namesake" is used, is...
(E) All of the above.

Examples:

(a) Person A was named after Person B
Baby points up at grandpa and sez, "Namesake!".
(b) Person B was named after Person A
Grandpa points down at baby and shouts, "Namesake!".
(c) Both (a) and (b) equally likely
Definitions extend beyond words and dictionaries and are augmented by situations, so sure, why quibble, equal.
(d) None of the above [please provide your interpretation in this case ]
A B and C refer to folks only. The word "namesake" can be applied and understood as a geographic place, or for that matter, anything with a name. So A, B and C can't be correct until you add an (E) All above are correct, then all become correct. I think. - -

Like in this sentence...
" Birmingham, Alabama is the namesake of Birmingham, England."
In this case it doesn't matter which name came first, I've heard it both ways.

Cheerio,
Milo.