A classical tourbillon rotates on a single axis: it compensates for gravity in one plane only, the vertical one when the wrist is lowered. Everything that came after stems from attempts to extend that compensation to the other orientations a wristwatch assumes throughout the day.
The construction principle of multi-axis tourbillons is cage within cage. In the bi-axial design, a second external cage is added that rotates on an axis orthogonal to the first: the two rotations combine and the escapement travels through all positions in space over time. In the tri-axial design, a third ring is added, with a third axis offset relative to the other two. The rotation speeds are deliberately different on each axis — in Eric Coudray's Spherion, later used in the Purnell Escape II, they are 8, 16 and 30 seconds; in Jaeger-LeCoultre's Gyrotourbillon the first two-axis implementations preceded the three-axis ones — because synchronized rotations would create privileged orientations instead of distributing them uniformly.
Each additional axis has a real cost. Transmitting energy through concentric cages introduces friction and inertia at every level. In Marco Lang's Seven Spheres, with seven titanium rings nested and offset by 30° relative to each other, the outer ring takes an hour to complete one rotation: energy must pass through seven gear stages before reaching the escapement. This demonstrates that beyond a certain number of axes, the theoretical gain in gravitational compensation is almost impossible to measure in practice, and the design becomes more an exercise in extreme micromechanics than a tool for improving timekeeping.
From the workbench I often see confusion between the number of axes and the number of cages or the number of tourbillons in the same movement. These are distinct things. A double tourbillon has two separate escapements; a bi-axial tourbillon has one escapement that rotates on two axes. Some movements combine both, like certain Jacob & Co. pieces, and there the brands' communication doesn't help clarity.
The gimbal mechanism in automatic winders applies the same principle to an entirely different purpose: two suspension axes, borrowed from marine chronometers, allow the internal rotor to remain always in optimal position regardless of the winder's angle. The tourbillon has nothing to do with it, but the geometry is the same.