Interesting comments. Much written above seems fine by me. I might describe things differently, but then that's my geek biomechanical perspective.
I so used to have fixed opinions on this. But I've learned over time that there are a number of different "valid" ways to think about this, depending upon how you view a turn.
This reminds me of a discussion where someone posted a Slovenian karateka with a home-made thingie (something like a Wing Chun dummy) that he was doing Uechi stuff to. Someone critiqued his moves as being all wrong, because most of his strikes were to the midsection of his gadget. But then Marcus brought up the point (from McCarthy) that a middle area from your perspective may be a head shot from your enemy's perspective - because you dropped them down onto their knees. Oooooohhhhhhh!
Same with the turn here. In the eyes of some, there's this right/wrong thing about whether you turn by "looking" first, or turn with the nose always facing the navel. I actually used to participate in those discussions, using a mantis as an example in nature where head often would look first while body remained in a Sanchin-like (mantis) posture.
And then I started doing aikido. And then I suddenly got the paradigm shift. Sometimes a turn or a "tenshin" isn't about facing a brand new bad guy from another direction. Sometimes the most unusual turns (like that 225-degree turn in Sanseiryu) are about getting another angle at the same person. As such, maybe you want to think along the lines that the fight is happening all the way through the turn.
Sometimes you feel like a nut, sometimes you don't.
So what I've found is if I talk with the student about their feet, or refer to the feet "starting" the turn - their bodies disconnect. Has anyone else found this to be true?
I could see how this language would get undesireable result. Certainly I'm not into opening up my femoral area for the world to inspect - unless I want a cardiac cath or something...
My language would be something in-between how Jim and Rick describe it. I use the back foot becoming new front foot analogy that Jim uses. I also like to freeze the turn halfway through like Rick does above. And when doing that, then I show how a half-way position actually can be such that you don't know which way a person is going if you freeze it. That's different than what Rick is showing above, where he clearly has the idea that he's going in a new direction - probably facing a new bad guy. Not right or wrong, but just is.
I like the concept of thinking of the leg and the hips moving as a unit here. I like to demonstrate this by going back and forth from start to halfway. At the halfway point, the navel has turned 90-degrees, and you have two front feet in two different directions.
Another point I like to make is this. In all Uechi kata turns, one foot always is firmly
planted on the floor while the other is pivoting. You do the shift of roles at the halfway point. This is REALLY important if you consider turns to be the drivers of technique. As your hips are connected to the feet, so too is the upper body connected to the hips. So basically you can think of the driving (planted) leg sending energy through the hips to the arms and hands for ... whatever.
And when it comes from foot to hips to hand, I don't see this as sequential summation of movement. For the most part, I'm doing simultaneous movement. If I choose to let the hands get ahead as Rick has done above (which isn't necessarily wrong given a particular application), then I know I have disengaged the power of my legs from my arms. Thus my arms are just getting ahead to touch or receive something. But disengaged in that way, there can be no action/reaction between hand and floor until the turn finishes and stance settles. These days, I hate to think of gaps in my kata. I like instead to think of energy and thus technique potentially (not always) being constant and unending.
But sometimes you want to whip. And then you no longer feel like a nut.