Great question. As far as I know, PSIA does not promote any particular "one size fits all" turn initiation for people skiing parallel.
PSIA documents have in the past talked about rotating the skis when in a wedge, I think. This is a vague memory, so I can't document this initiation strategy as the "one" promoted by PSIA. I do know that many trainers talk about starting a turn in a wedge by rotating the skis, rotating the toes to point in the turn's direction, rotating the femurs, or pointing the arrow in the direction of the turn. Take that onto blue terrain with a new parallel skier who isn't interested in taking more lessons and what do you get? Conclusion: we need more than that one turn strategy.
When I took my LII exams, "extend-to-release" was a popular phrase, so I used it. I passed. My interpretation of "extend-to-release" is to lengthen the new outside leg, starting with its little toe edge engaged. I did not get that from any PSIA document, however. I did hear clinicians talking about extend-to-release for several years, usually without any explanation of what it meant specifically. Training is so very different one ski school to another.
In a more recent Level III prep clinic for the skiing exam, our clinic leader, who served as a selector the the National Team last time around, said a "retraction turn" was what PSIA wanted to see us doing in the exam. I expect what he meant was "flex-to-release" ... shorten the new inside leg to start the turn. But he did not elaborate. My understanding of "retraction turn" is quite specific, involving bringing both feet/skis up under the body and putting them down on edge on the other side, prompting the skis to offer rebound. I am fairly sure he was not talking about that as the go-to turn initiation for personal skiing.
Seems like if LIII skiers are supposed to flex-to-release, keeping their bodies low between turns, in their personal skiing, that should mean that teaching advanced intermediates to flex the new inside leg to start a turn (instead of rotating both skis, or extending the new outside leg, or tipping the new outside ski onto its big toe edge, or whatever) would be the promoted PSIA initiation. You ask about ankle/legs. I suspect he meant by "retraction turn" a leg flexing movement, not a foot movement, but that's me reading into it. If this is what he and the other selectors were looking for in the candidates for the Demo Team, I still doubt PSIA is ever going to put that in writing for the general population of ski instructors.
When I read the last Alpine Manual a couple of years ago, I looked especially hard for any description of how to initiate a turn. Any turn. Couldn't find any. I looked again last summer to see if I missed it. I couldn't find anything on that second look. Maybe it's there and I keep passing it over. Don't think so, though.
I've been wanting PSIA to list different ways of initiation turns and when/where they might best be used, but when I approach people who seem to be in positions of power or influence, they don't want to do it. BERP and the new Five Fundamentals are structured to avoid saying anything specific like how to initiate a turn. This is on purpose. It's because PSIA's philosophical approach to its educational materials, for some reason, is to write "principles" that will apply to all turns, on all terrain, in all conditions, using all gear. In other words, they intentionally water everything down to broad generalities. This means they intentionally avoid discussing what to do at the start, middle, and finish of a turn on a groomer, in bumps, in a narrow couloir with yesterday's now-crusty snow, etc.
From a past Demo Team member this last April, I heard that the reason our Demo Team does a not-so-good job at Interski with its synchro-skiing is because of this resistance to choosing one turn mechanism to use. Austria, on the other hand, has all its DTeam members skiing the exact same turn in those performances. Those skiers are all on the same skis, too. Their synchro skiing is exemplary. Ours ... not. Our skiers are on different skis, using their favorite movement patterns. Matching what each other does can't work well given those facts. Diversity doesn't look so hot in synchro-skiing.
So probably the USA's embracing of diversity is at the heart of the issue. I can't see that embracing diversity necessitates avoiding talking about different initiation strategies, though. Maybe it's just politics; the old guard vs the new, and the philosophy masks that conflict.
If someone knows I'm wrong about there being no PSIA guide delineating how to initiate any particular type of turn, please provide a link. I'd be very interested.