Here the two elemental schemes are furthest apart. Japanese esoteric Buddhism makes the middle finger fire, ka, the third of the five elements counted inward from the little finger and the one at the centre of the hand. The yogic hasta mudras make it ether, akasha, the subtlest of the five and the one the others are said to arise within. Fire and ether are not neighbours in either cosmology, so this is not a small slippage in a shared list. It is evidence that the two schemes were built independently and that neither is derived from the other.
Both do agree in giving the central finger a distinguished element rather than an ordinary one, and that is probably the anatomy speaking. The middle finger is the longest and the most nearly axial: the hand closes around it, the other digits fold toward it, and a fist is built on it. A scheme that assigns five elements to five fingers has to decide what the middle of a hand is, and both of these decided it was something central rather than something common.
Its best known modern use is an insult, and the gesture is old, attested in the Roman world long before either elemental scheme reached the form it now has. That is worth stating plainly beside the ritual material, because a finger can carry a sacred meaning in one tradition and an obscene one in another without either being a corruption of the other.
Sources Mudra: A Study of Symbolic Gestures in Japanese Buddhist SculptureE. Dale Saunders with Asana Pranayama Mudra BandhaSwami Satyananda Saraswati