Thanks for starting a new thread and a fun subject to discuss, but I think you're missing the most important part, the arm that's above the surface, recovery arm. I think we make the mistake of separating movements/parts into different depts or somehow all are mutually exclusive - and that's largely due to the traditional language of one arm catches and pulls or "pulling arm" and the other recovers (until the next pull) or "recovery arm", and legs kick. This implies one is active below the surface (catch/pull) while the other is inactive or resting above the surface (recovery).
I use the language neutral, "high side" and "low side" arms (Boomer, Kredich), neither imply action or rest. And I view what happens above the surface (in free/back/fly) directly affects and *connects* all movements below the surface.