I used to use a full cheek snaffle. Then I switched to a bitless bridle. I was surprised to learn that my horse is much more responsive in the bitless - we both like ti better.
There is only one exception - speed competitions. We do mounted shooting and my horse needs something in his mouth for that. For that I use something that looks exactly like the second picture in smrobs post above.
It is tricky since my horse is very sensitive to what is in his mouth. The bitless bridle can make you get sloppy hands. When we use the bit, I have to concentrate on VERY subtle hand movements. Too much hand or a tight rein and my horse will react poorly. Tough to do when racing through a course.
We definitely practice with different bits.