I did a trip a few weeks ago that involved crossing the Gila river over X100 in the course of 4 days, most crossings were mid thigh to navel high and relatively swift snow melt. I had done crossings in the past but none that tested me physically like that trip. Like you said, there are a lot of different ways to do so, there are probably even traditional techniques taught in outdoor leadership schools and trekking courses. I've never been to any of those, more baptism by fire than anything.
This is only my experience but what I've found to work is:
Face across the stream and proceed forward, get a feel for the current and depth and check your footing before committing with each step. I use trekking poles and these serve as an invaluable tool for checking the rivers bottom ahead of me for depressions and debris that might take me down, they also can be used to "tripod" your body in the water. Placing one pole in the water at an angle on the rivers bottom downstream alleviated some of the force of the water and on my body. I wouldn't have been able to cross the river safely without the use of the poles IMO, they helped provide 3 points of contact with the rivers bottom when one foot was suspended in the water advancing forward. I've found that as the river gets more swift that I tend to slightly increase my lean forward and angle my body upstream gradually. One problem I found was sometimes I would lean too far forward facing upstream and on two occasions the water currents at the rivers bottom knocked my feet out from under me, this was mostly due to my center of balance being too far over my feet. But since I was facing upstream I was quickly able to brake in the water and plant my feet, my pack stayed dry on both occasions but our digital camera in my hipbelt pocket didn't! I did find that when I was fearful of the crossing or nervous I would get chicken legs in the river, usually I would back out of the water, regain my composure and try again with confidence. Basically, don't commit unless you're ready.
There is this weird affect I experienced on a couple of crossings where I would slightly panic and be forced into a standstill in the middle of the river, looking upstream I lost my equilibrium watching the water rush past me, for some reason the water passing by me made me feel dizzy, disoriented and out of balance. I would have to snap myself out of it on a few occasions and keep moving forward. Anyone else had this experience?