Thank you all for your comments. I also felt that the precise statements from the coach were a little dogmatic, but I also think, as he has coached many people, that his comments must have has some solid empirical basis. That is, he probably noticed that the vast majority of swimmers are short of air by breathing every three strokes, and that the 2bk almost never works, in his experince.
But this kind of feedback is why I like reading this forum. It is always interesting to hear the personal experience from swimmers at different levels, and having different and sometimes contrasting opinions helps me to avoid being too dogmatic.
I agree with Pat though on the fact that relaxed might not mean efficient, especially before the correct movements are imprinted. I found that when I have to correct some parts of my form, relaxation only comes second, after correcting the form. For example, I had a ‘relaxed’ kick, but I found, after filming, that I was kicking too much from my knees, not from my thighs. Changing this was not effortless… Same for having a long, extended form, I had to stretch myself, and only when I adapted and became flexible, I could relax into the correct posture.
I will follow your suggestions and try to make the 2bk more natural. I also feel the gliding phase of the stroke more. Also, I think ames makes a good point, in that the 2bk may emphasizes the problems in the stroke, and not necessarily cause them.