It's highly subjective but I would say the functional distinction between "cup" and "dish," especially for Danbooru purposes, is whether they are used for drinking or eating, not the geometric shape and what (Western) dining utensils that shape is associated with.
Imagine someone searches for "cup". They're probably not looking for people throwing beans from a masu, but they may be interested in someone drinking from a sakazuki.
Likewise, imagine someone searches for "dish". They're probably not looking for people drinking from a sakazuki.
Also, Wikipedia calls sakazuki cups: https://en.wikipedia.org/wiki/Sake_set#Drinking_cups
