Help me sort out the PythonCaller syntax from where AI left me to fend for myself

Is this everything that is in your pythoncaller? Since your are using the AI in the pythoncaller I take it that your FME version is probably 2024+? In that case you need to use classes instead of just functions. I cant test if it works but I would expect something like this:

import fme
import fmeobjects
import math
import numpy as np


class FeatureProcessor(object):


    def __init__(self):

        pass

    def input(self, feature):
        angles_deg = fme_feature.getAttribute('angles_deg')
        deviation_rad = circular_standard_deviation(angles_deg)
        fme_feature.setAttribute('circular_standard_deviation', deviation_rad)
        self.pyoutput(feature)

    def circular_standard_deviation(angles_deg):
        angles_rad = np.deg2rad(angles_deg)
        sin_sum = np.sum(np.sin(angles_rad))
        cos_sum = np.sum(np.cos(angles_rad))
        mean_angle = math.atan2(sin_sum, cos_sum)
        R = np.sqrt(sin_sum**2 + cos_sum**2) / len(angles_deg)
        deviation_rad = np.sqrt(-2 * np.log(R))
        return deviation_rad

Do note that I haven't actually taken a look at the functionality of the provided code, I did some copy pasting into the format that FME wants. 

Adding to what ​@birgit said, but specifically for your last question: “...what should I write in the box: Class to process features?”
The Class to Process Features should be whatever is listed as your defined class. That first object you define that you want to process. In this case it would be “FeatureProcessor”. When using the PythonCreator the default is FeatureCreator and for the PythonCaller it is FeatureProcessor.

Thanks all!

I got it working with the pointers above and some more digging in the documentation.

(I should ad that the final script does not look much like the one in my example, so don't count on the code above to be useful.)