Recently my students and I began a discussion about the proper way to execute a fouette turns en dehors. It started me on some research that has proven to be fascinating. According to ABT's online dictionary, the Russian method extends the leg to second position before whipping in with a strong beat of the foot back to front. The Ceccetti method extends the leg toward the front first before begin carried side and pulled in for the turn, while the foot moves from the side of the knee to the front. What is interesting is that of the many Russian trained dancers online, I could find none that truly open to the side only before being pulled in. And, I saw very few that didn't do a pronounced beat around the leg. As I was telling my students, I think this makes sense because of the nature of the more extreme technique we see on stage today. To pull in to 3 and 4 pirouettes from the fouette requires a more vigorous whipping of the leg from front to side. Having said that, what also is apparent is that fouette turns, and turns in general are going to be slightly different for every dancer. So, while there is a gold standard for what to aim for, the reality is that every dancer may use his/her arms in slightly different ways, have small differences in preparations and in timing. See below a few examples.
The first video is from Royal Ballet on the history of the fouette:
In this video, please observe the more subtle transition of the foot and the more extreme, angular arms:
Lastly, look at fouettes that I personally think are beautiful. I like the floaty feeling at the height of the turn and the arms which aren't extremely exaggerated.