Dave Arnold showed in 2009 how the technique of shaking doesn't matter after 12 second given that the shake is vigourously hard.
What is missing is experiments on technique and how it forms foam in cocktails.
For one, we can't easily measure foam density. Though we can measure foam volume, it doesn't say much about the density if we can't measure its mass. It is very difficult to because foam is mostly air.
As such we can't consistently test actual tecnique against each other.
We can however guess that the best technique to create dense foam is a shake that applies force parallel to the liquid's surface. With large ice
We know from GI Taylor's The Four-Roll Mill experiments extensional flow (stretching) shatters bubbles far more efficiently than shear flow (tumbling).
By shaking parallel with a large ice cube, you force the liquid through extremely tight bottlenecks. This creates Taylor's stretching force, rapidly snapping large air pockets into the tiny micro-bubbles required for a dense, stable foam.
The technique that allows for the densest foam with least amount of effort is a shake with the tins parallel with the floor surface with a strict horizontal piston action. Shaken with enough force to snap large bubbles into smaller micro-bubbles.
A shake up and down is wasted fighting gravity on the upward motion. While a shake that changes the angle (looking at the 3-point shake) defects extensional force that causes bubbles.
A note about 3-point is that the because the angle changes, it actually bounces ice in the tin a lot more. This chips ice in the tin more than a strict piston movement. The result is a unique texture in gimlets and daiquiri where is is almost a blended style texture with ice shards. Although, in terms of dense even foam, 3-point shake is not the most effective
