The term "shepherd's pie" did not appear until 1877, and since then it has been used synonymously with "cottage pie", regardless of whether the principal ingredient was beef or mutton.
The Wikipedia article also supports this claim with links to seven different sources. The quiz is correct.
"The term shepherd's pie did not appear until 1854, and was initially used synonymously with cottage pie, regardless of whether the meat was beef or mutton. However, in the UK since the 20th century, the term shepherd's pie is used only when the meat is lamb."
Three links verifying the latter claim. I guess outside the UK it may still be synonymous.
AAHHHH!. Hot potato, orchestra stalls, Puck will make amends!