OC 181 (4) MB  says that when washing mayim acharonim one must wash up to the second knuckle and using a few drops does not fulfill the mitzvah at all. Shulchan Oruch HaRav 181 (4) says there is no shiur for the water used but agrees about the second knuckle. Where does the custom to wash the tips of one's fingers with a few drops come from?
See this OU article on Mayim Achronim:
A bit more of an explanation from here (post #6):
Either way, we see clearly that this "custom" was also around in the times of the Mishnah Berurah (and perhaps even earlier)...
Indeed, see the OU's article for more sources for how it's not correct...
If not for seeing the above article, I would've ventured a guess that this "custom" stemmed from how one cup of water for mayim achronim is generally used for a somewhat large group of people (not necessarily purposely, but that's just how it seems to be done). If each of those people would indeed pour a large amount, the water would very quickly be emptied.