Crashing on ice is just about the definition of an inside corner crash. You are highly unlikely to go over the high side when you crash on ice because you don't have much momentum, there's nothing to catch the wheel and you aren't flying through the air. Every ice crash I've experienced is exactly like the crashes in first 2 videos. The bike goes down first and takes the rider with it. Trying to put your foot out does no good whatsoever and, as hairytoes points out, results in you and the bike going one way and the leg you put out going in a very uncomfortable other direction.

Let's look at the physics of crashes on ice: The rear wheel slips out to one side which, in itself, isn't a problem. The front wheel loses traction which is a huge problem. You, as the rider, can't countersteer to get your bike upright again. It's inevitable that you are going to hit the ground. Now the fun stuff happens.

If you try to save yourself by throwing out a body part, that body part is going to hit the ground first with the full momentum of the bicycle, rider and any extra weight the system happens to be carrying. If you throw out your leg, the bicycle and your body are moving in one direction while your leg is moving in another direction. If you like being the wishbone at Thanksgiving, this will be your moment. But, just like the wishbone, something is likely to break. Even the most limber person is probably not going to be able to do the splits without some preparation and cold isn't conducive to limber muscles, joints and ligaments. The best course to keep your legs inside the bicycle.

If you throw out an arm, your arm from wrist to shoulder to neck becomes the focus of an awful lot of load. If you don't break your wrist, you could very easily transmit that force up the arm to break any one of several bones along the way up to the collar. The human collar bone is a terribly fragile bone and pretty painful to break and slow to heal. You could also tear a whole host of muscles and ligaments. You could pull your arms in but now your shoulder...the bit you fear hurting...is in danger.

Now say that you could jump clear of the bike. What are you planning on doing to avoid hitting your other bits on the ground. You've got momentum, you've got a nearly frictionless surface, you've got a biped sliding on ice on two (or one) feet. You are likely to end up on the ground in a heap anyway. You've almost guaranteed that some part of your body is going to make a painful impact with the ground. Which part? Quick, you don't have a lot of time to decide. Take it on the coccyx...even more painful than a collar bone. Take it at the shoulder...wait, that's what you are afraid of. Too late. Something hit and it's gonna hurt.

Again, say you could jump clear of the bike and not manage to hurt yourself. You are sliding like a seal on an iceberg. The bike is sliding but the bike has all kind of protuberances that are slowing it down. You are sliding in the same direction but the bike is sliding a little slower than you are. Do the math. What hits what?

Let's say your get clear of the bike. You are still a seal on an iceberg. If you happen to hit something, you hit it. Just like the fourth guy (about 20 seconds) in the Innerleithen video who just about creams the tree. Most of the other riders crashing in the video let the bike take the hit to the tree. He tries to save himself and almost becomes a limb for his efforts.

Now look what happens if you stay with the bike: Some muscle mass is going to hit the ground in either case. But the bike becomes an extension of your skeleton and the handlebars, frame and bags take a lot of the impact. By being loose during the fall, your connective tissues take less of a beating. By hanging onto the bars, you know where the bike is and it doesn't become a separate missile that you can hit or will hit you. The bars also let you leverage yourself away from the bike so that that all important shoulder joint doesn't get damaged.