Because Swingers use the Sequenced Release whereas Hitters use the Simultaneous Release, it is makes sense to fan your forearm gradually all the way to the top for a Hitter, since this mirrors the motion of downstroke release for a Hitter. Likewise for Swingers- most of the fanning is completed at the startup since this is more "compatible" with the Sequenced Release.
You do not want to use Push Stroke for anything more than a chip. It is best reserved for putting and chipping because of its lack of power. A lot of effort produces only small changes in distance for the Push Stroke, so it is an ideal stroke to use around the greens.
From what I know, you can use Right Forearm Pickup (with Extensor Action) for with Hitting and Swinging, therefore this "technique" should have a separate identity to fanning.