your questions are good ones. I'll do my best.
Generally spoken - if the opponent is close you'd use cutting techniques, if he's in greater distance you'd use thrusting, striking or chopping techniques.
What I term as "shao" is "xiao" (with knife-radical on the right side) not sao, since the name doesn't refer to the arm movement. The meaning is "to peel away" and it means to peel away the flesh of the arm or the shoulder from the bones with a quick movement, initiated from the power of your back. Thereore in shao/xiao you have to use the very sharp qianren-zone (first third) of the swords blade. Shao/xiao is very frequently used and it apperas in many variations, e.g. also as a stroke to neck or ear, but always in palm up grip. It's generally used after a defensive technique as ya or dang. Xiao is often used in ping-pong to describe a cutting-slice with the backhand.
With YZD's Phoenix spread wings I have a problem, since I now realize for the first time that he has Phoenix spreads right wing and another Phoenix spreads left wing. I can say nothing to this, but Phoenix spreads right wing is shao/xiao. Phoenix spreads left wing to me is a Little Dipper (Ti) followed by what is called "Whirlwind" or "Tornado" in slow pace (initiated by ya).
YZD's "jiao" is what I call "ge". I use two different terms for stirring, since there are two different stirrings. My jiao is stirring while the sword is in the body of the opponent to make the wound larger and to increase the pain. In traditional Yang sword form there's a second thrust after Cat catches rat before Dragonfly touches water. Between these two thrust my "jiao" is shown.
YZD doesn't show this second thrust, therefore he doesn't have a special term for this technique.
YZD's jiao/ge is a technique in which the sword is held horizontally or vertically and shaked, so the blade moves in one or more circles around the opponents weapon, either from inside or from the outside. Whith this jiao you try to force the opponent to drop his weapon or to make him loose his balance.
It's a very difficult and dangerous technique, since f you fail you will easily loose your own balance. Applying more than one circle needs a high degree of sensivity. I cannot detect this jiao either in YZD's nor in Yang Yun's form. During the single technique demonstrations on his video, YZD really shows a kind of jiao at the end of Wind blows Lotus, just as you mentio0ned. After he carried the sword from left to right with palm up, he circles it one time aas a transition into the first Lion shakes head - jie (palm-diagonally down). Unfortunately the video has a kind of jump here, so I cannot see 100% whats going on. But Yang Jun doesn't show this in the following demonstration.
Jiao/ge is always possible when your sword has contact to the opponents weapon. Therefore you could have many jiao/ge in your form, particularly during grip-changing positions. Therefore you can surely have a jiao in Dragon walks left/right. In this position the opponent retreats, while you walk forward unstoppable as a Dragon while thrusting to the opponent from left and right. First the opponents sword is neutralized to the right (palm down). Then you jiao in Palm-up grip and thrust forward.
This jiao also helps to make your thrust in spiral-form, not straight.
In Sword form it's also desired to have spiral thrusts instead of straight thrusts. This aspect plays a part in the transition from Little Dipper to Swallow enters net.
First the sword is hold in that way to keep balance during the 180 degrees shift. The tip points to the left, although you want to thrust straight out. To do this your swordtip has to move in a spiral. If you'd carry the sword by your right side, you'd thrust out straight. In YZD's form this is not so obvious, since in Little Dipper he makes Ti with the xiaren-edge (finger side edge) of the sword (palm down). I do it with the shangren-edge (thumb-side edge-palm up).
Then I step back, lower down and thrust in a obvious spiral forward, coming up as from Snake creeps down.
Tiger holds head is an own position but it also precedes Cat catches rat. It's shown in photos 28 and 29 in YZD's book. The sword is
hold up as if you'd read sometghing from it's blade - for stabbing the opponent's throat) or for blocking something from the front. Cat catces rat starts with sinking the sword and the right foot for the jumps. The point in Cat catches rat is to puzzle the opponent what's your real attack. Your sword goes down, up, down, up and finally down. He is retreating - his sweapon is where it is. You can chase him for many jumps, also you can wait with the finakl thrust in your right bow stance, while shifting weight back and forth and mowing blade up and down as seen often by fencers during olympic competitions.
Jie and Ji have different grips. For Ji palm faces up 45 degrees, for jie it faces down 45 degrees. Ji and Jie are done with the arm in the beginning. At the end, the wrist has to be stretched to its maximum to make the cut larger. That's also important for pi. Since shao/xiao is done with the qianren,
ji, jie or pi uses the second blade third, the zhongren, which isn't so sharp but more stable. The last kick of the chop is made by the wrist-strech, using the qianren again.
Only choping or striking down, without wrist flex is not correct.