I like this thread because it has made me think a little.
First, I think Lakota is absolutely correct on how to TRAIN an untrained horse to yield to pressure. You are in a controlled and good learning environment, and you teach them to move as described. It is timing, feel, pressure and release (particularly release) that trains.
However, I didn't read this as a "first training" opportunity. This is like my kids when I say, "If you don't go to bed by the time I count to three...." I always end up counting to three. They don't move on one or two. If I counted to 100, they wouldn't move until 99. This horse seems to understand what you are asking, but is ignoring your request because she doesn't see any reason she should do what you want, rather than what she wants. Also, this doesn't sound like its happening when you have time to go back to the early basics. You need to get results quickly so you don't get hurt.
This then, is why I think a hoof pick or baling wire works, and works better than a crop.
First, I would explain my "ask, tell, demand" comment. I think a problem with many people when they train their horses is they ask, and ask, and ask the horse. There is no escalation, no reasonable consequences for the horse's actions, simply more and more asking. It goes to my "counting to three" example. We tell the horse "move over, please". When the horse doesn't move, we ask again, and again. Why would the horse ever move in this situation? We are effectively training the horse to ignore our instructions. So, first you ask (and reward the littlest try), then you tell the horse (now I'm getting right in there and trying to "show" the horse what I want), then I demand the horse move.
If my horse is tied at the wash rack, and I want him to move his hip over, I will cluck at him as I walk up. For my horses, this starts them knowing I want them to 1) listen to me; and 2) do something. I'm watching as I cluck for the smallest try to move where I want them to move. If the horse moves, I stop pressuring and release. Cluck and see if they move again. If not, I increase the cluck rate. If there is no try, my cluck now is pretty fast, and I will touch the hip where I want movement (again, asking for the horse to move away from my pressure). Release for slightest try. Touch again for another try. Finally, if clucking and touch get nothing. I will push on the hip with the top or back of a hoof pick, increasing the pressure until the horse steps away.
There are two reasons why I like the pick or wire in this training scenario, better than a whip. First, the horse doesn't really see why you can push so "hard" (or so effectively) with a pick or wire, so he can't "anticipate" you upping the ante (like if you have to reach over and get a whip). So the three step escalation is very easy to do, and is relatively invisible to the horse. First you cluck/will/move-with-feel to get the horse to move; then you cluck/touch to get the horse to move; then you cluck, PUSH with pick so the horse must move. Secondly, unlike with a whip, with a pick or wire release is instantaneous when the horse responds appropriately. It doesn't really take feel. You press with pick, horse steps away and gains release.
The other thing I would add, for what it is worth, is that I would make this a "non-event" emotionally. I would not get "big" or put a lot of energy into moving the horse. You want a horse that moves when you cluck (the first level of command), with the lowest "input" of energy. You don't want a horse that waits until you escalate to being big and animated. So I would keep this very low-key for training. I cluck, touch, push with pick without making a big deal. Pretty soon the horse is moving away just when I click.
This also speaks to Cherie's comments about "pecking" on a horse. If you don't increase the ask (and consequences for not responding) then you are just "pecking" (asking over and over).
Now I've never been one for generalizations, and I will comment on Lakota's statement that horses "always move away from pressure." I would say they move away from pressure if it is in their best interest. I have watched a horse l-e-a-n into a barbed wire fence so hard it bloodied it's chest just to reach grass on the far side. Clearly that horse decided to move into pressure because the perceived reward outweighed the penalty. As an Appy owner, I know they will move into pressure aggressively rather than pull, and in my experience some other individual horses will move forward unless trained otherwise. It also doesn't work with mules, they go into pressure.
Thanks for making me think about this.