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:
According to the kabbalistic interpretation of the Kaf HaChayim Soffer, mayim achronim is an “offering” to the sitra achra—the “other side” —and therefore, must be removed. (This notion of the sitra achra is also mentioned by Rav Palache in the name of the Yalkut Reuvani, who states that Iyov (Job) suffered because he neglected to perform mayim achronim.)
Interestingly, the concept of the sitra achra also serves as the basis for people washing only the tips of their fingers. It should be noted, however, that according to most authorities, washing only the fingertips is insufficient to properly clean one’s hands prior to saying a brachah.
A bit more of an explanation from here (post #6):
The reason they use very little water is because of the reason cited in the Ben Ish Chai that I quoted before - that Mayim Achronim water is tied to the [Sitra Achra] and therefore we should produce as little of it as possible.
Either way, we see clearly that this "custom" was also around in the times of the Mishnah Berurah (and perhaps even earlier)...
The Mishnah Berurah (181:10) decries those who, while careful to wash, use only a few drops, leaving the hands insufficiently clean for bentching and thereby not truly fulfilling their obligation.
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.
Ben Ish Hai Shelah #8
יזהר שיהיו מים האחרונים מעט ולא ירבה בהם כי המה חלק הסט"א