I have an engineering background so I enjoy these kinds of discussions, or maybe I went into engineering because I like these kinds of things, who knows? I find it really interesting the variety of ideas that come out and the difference of opinions. Some people have what I think is a pretty good grasp of things and other not so much (but think they do) so it can make for some interesting dialog! I've fallen into both camps myself. I've explained things to others and in turn have gotten schooled on things too. While I think I have a fairly good understanding of the mechanics and physics behind steering a motorcycle, putting it into words is a whole other story. I'll at least make an attempt to share my point of view (right or wrong) by addressing some of the points brought up in the thread so far, in no particular order other than the order in which I think of them.
First off, the gyroscopic effect. It has been stated that it plays a major part, and that it plays no part what so ever. I think the truth falls into the middle. The gyroscopic effect plays a primary roll in the stability of the bike; the faster you go the more stable it is. Try riding without any hands on the handle bars at 10 mph, and then try at 50. The bike can wander all over at 10 mph, but stays fairly true to line at 50 mph. Also look at stunts that super cross racers do; the one in particular where they kick the back of dirt bike out about 90 degrees and then bring it back straight before landing. They can do that by manipulating the gyroscopic effect (as mentioned before with the 90 degree reaction force bit). The gyroscopic effect does not cause a bike to turn (unless you're airborne), quite the opposite, it resists the change in direction (hence giving the bike straight line stability at speed and at a given lean angle when cornering). Another way to practically see this in the real world is to weave between the dotted lane marker lines. Try it at 30 mph and then at 60 mph. Try to keep the tire close to the lines so you're side to side distance traveled is about the same. You'll find it takes a lot more input from the rider to weave back and forth rapidly at 60 mph then it does at 30 mph. I've done this myself and it can be hard to do it quick enough (even on a light weight dirt bike) at speed do you don't overshoot a line.
The contact patch quote the OP posted has been throwing some people off and it took me a few reads to figure out what he was saying. A couple of people have correctly explained it but I'll put it in my own words that may or may not help someone. I think the missing key information is that the "circumference" is not the circumference of the contact patch; it's the circumference of the TIRE rotating about the axle. The contact patch is basically a little disc and as you lean the bike that disc becomes angled relative to the axle. The side of the contact patch closer to the center line of the tire (outside of the turn) has father to travel in one revolution of the wheel than the side of the contact patch that is closer to the axle (inside of the turn). Because they are physically locked together, this causes a torque that try to turn the wheel about the axis of the steering head. That said, I think this has almost no affect on the bike actually turning.
Now, it is my opinion that the steering of the bike is almost (like 98%) entirely based on the geometry of the front suspension: axle relative to the center line of the steering head, or in other words rake, trail, and offset.
Here's a link to a site I found via a quick search on google that gives an explanation of the geometries. The shape of the tire also plays a roll in the geometry (like how the handling goes to pot when the sides of the front tire get cupped or flattened off). I'll try to explain. The geometry is all about gravity. When sitting up straight, the center line of the contact patch is in line vertically with the steering head axis so it is balanced and doesn't want to turn left or right until it is forcefully turned. Once it is turned the contact patch on the tire moves sideways away from the center line of the steering head because of the trail of the front wheel (see link above) and the wheel naturally wants to flop over to the side in the direction of the lean because the contact point of the tire creates a torque about the axis of the steering head. You can see this stationary too. Put the bike on the kick stand with the wheel straight. Is is easier to turn the handle bars to the left or to the right? Turning them to the left drops the bike closer to the ground whereas turning to the right lifts the bike up. Now you might ask why the front wheel doesn't just keep turning further and further on it's own when you're leaning around a corner, and that's because the centrifugal force counters gravity trying to pull the front wheel over in the direction of the turn; same reason you don't just fall over too. A properly designed front suspension geometry will be "neutral" meaning that at a equilibrium speed, lean angle, and turning radius of the road, the bike will maintain the turn without under or over steering on it's own. Motorcycle publications will talk about the rake angle and trail and talk about how quick a bike will "turn in" and this is what they're talking about. Generally a higher rake angle (more vertical) will be quicker to turn, but less stable at high speed where as a low rake angle (more horizontal) will be more stable at speed but not as nimble in the turns. When was the last time you saw a steering damper on a chopper!?
Now as far as counter steering goes, on a properly designed bike you ALWAYS counter steer; both to initiate a turn and to control it. The counter steer action initially turns the front wheel in the opposite direction that you want to travel to initiate the bike lean, and then as the bike leans the wheel will naturally turn into the direction of the turn without further input from the rider. As the rider wants to change the magnitude of the turn, you have to either initiate more lean or less lean by counter steering. As you're going around a corner, the bike rides in it's "neural" state. If you want to turn tighter, you need to lean more and to lean more you counter steer a little; that doesn't mean you "turn" to the right in a left hand turn, it just means you turn a little less left. You can try this by riding around a corner of your choice but keep your palms on the handle bars with open fits so you can only make corrections by pushing on one side or the other. Try to move from the inside of the lane to the outside and back. This really hit home with me when I was in high school tooling around on my dual sport. One day I was doing something that required me to pull my right hand off the throttle (can't remember what) but in order to keep riding I had to use my LEFT hand on the RIGHT handle bar to control the throttle. I was all over the road until I figured out that I had to push/pull in the opposite direction that I wanted to maintain control of the bike. I had been riding bicycles and dirt bikes for more than 10 years and I was doing everything by muscle memory without consciously realizing what I was doing to control my bike. It sounds like a lot of you guys are in the same boat I was. If you're really in for an adventure try crossing a hand over to the opposite handle bar, but be careful (and you didn't get the idea from me)!
I probably wanted to comment on something else too but now after writing all this I've forgotten, so I'll call it quits. I hope this helps someone.