Both from the same jump (a flip jump) and the direction of travel is basically about up the center of the screen caps (bottom to top), to give you an idea of the pivot action. Some skaters pivot more, some a little less. The jump can pivot to forwards and still be clean on the take-off. A lot of ladies skaters actually do this (Asada and Sotnikova, for example) on a lot of their jumps. Kim's flip is more similar to the one linked above - that take-off is actually conducive to a bigger jump. Some coaches call it a "power take-off." The fuller pivot sometimes work better for skaters who don't jump as high, but rotate well and want to get into their rotations ASAP.
The "full blade" jumping that people complain about (which only happens sometimes on badly tacked on doubles in combinations or by low level skaters, which are usually severely prerotated anyways i.e. toe axels), is nothing more than a screen cap of the skater's body absorbing the impact of the jump take-off. I can write a whole page on this, but I think the best way to see how ridiculous it sounds is to put on some figure skates and try to jump any decently sized double, triple, or quadruple jump off the skate rocker and not possibly give yourself a concussion after your skate flies forwards and you fall and bang your head on the ice.
You can pivot to forwards on practically any non axel jump and get no deduction or GOE fault. The judges won't even be bothered by it. They also won't think of it, because they know (many of them being former skaters themselves) what happens on a jump. There's a nice little saying among many coaches: "All jumps take-off forwards." It's true. The ENTRANCE EDGE may be backwards, but the actual TAKE-OFF of the jumps are more often than not closer to forwards than backwards.
On an axel you can go sideways or a smidgen past sideways.
There is no elite skater competing right now that I know with straight back and forward jump take offs. Not on triples and quads (or double axels). Most probably wouldn't consider it safe to jump that way. I don't personally know any coaches that teach jumping that way. Even back in the 1940s when Dick Button was doing Triple Loops for the first time, he pivoted to forwars on his take-off and many of those skaters had the same reaction with the ice in their skate/ankles on jump take-offs, because to try to stop that is to willfully risk injury (Ankle Sprains, Back Issues, Knee Issues, possibly even broken bones, and other types of overuse injuries).
Well, several things:
1. Seems like Yuna and Polina at least (I haven't looked at many of the skaters in detail) will do their lutzes and flips taking off still facing backward. As in, both the edge foot and the toe pick foot are still facing backward when they lift off the ice (as in, facing the backward half). While Mao, Adelina, Gracie, and probably others will do their lutzes and flips such that the edge foot lifts off still facing the backward half, but the toe pick foot will have rotated to be facing nearly directly forward by the time it leaves the ice. I'm puzzled by this. Certainly, on some jumps (like the salchow), pretty much everybody does the edge going backward, then pivot on toe pick until facing nearly directly forward before liftoff thing. But are the coaches for the former skaters so sadistic as to make them do an extra quarter or slightly more rotation in the air compared with the latter skaters, if points-wise there is no difference? Is there no advantage to taking off with the toe pick still facing in the backward half (rather than take the whole allowance and be facing nearly directly forward) at liftoff, either in terms of technique or in terms of points? If so, why do some skaters do it? Yes Trevor's site mentions that maybe coaches don't know about it, but these are the Olympics; I find it somewhat hard to believe that Olympic-caliber coaches don't know about these things. For example, in this thread, one of the points of discussion is whether or not it counts for anything that Yuna hardly pre-rotates and is still facing backward when she lifts off, in terms of how much she has to rotate in the air and the allowed under-rotation due to that. Again, I would find it very odd for her coach to tell her "yes I want you to do an extra quarter revolution compared to many other skaters" if it leads to possible under-rotations and points deductions for them. Why put in the extra effort?
2. I know the term has been mentioned a few times, but I'm confused as to what the "power take-off" thing (as opposed to a regular take-off) really means. Is it pivoting less on the toe pick for more power/height or something? I don't know if that will explain #1.
3. My question with full blade jumps is more about when the skater puts the full blade down for the toe pick foot, not the edge foot. Some skaters do it, some don't, but I don't know if there's any penalty for doing so. If not, why not? If so, where is it in the rules?