Looking at the equation under the title My calculation, you have defined a quantity sigma equals R * delta_theta.

In this equation, sigma is the length of the circumferential path traveled by a point on a circle of radius R for a rigid body rotation of delta_theta.

That equation is correct. Then you rearrange the equation to compute delta_theta from sigma/R.

This is also correct assuming sigma is measured using the definition above.

But how did you extract sigma from the FE results?

It would be wrong to request the Total Deformation of a node on the edge of a circle. For example, say you had a circular rod with a diameter of 10 mm. Now twist that rod 180 degrees using a Remote Point (Behavior Rigid). A node on the edge has moved around the circle to the opposite side of the circle, but the Total Deformation result shows 10 mm, even though the path it traveled was pi*5 = 15.7 mm in length. If you twisted the rod 360 degrees, the Total Deformation result for that node would now show 0 mm though it had traveled 31.4 mm.

You show a Remote Point and a Flexible Rotation Probe. That will correctly measure the rotation if used appropriately.