I almost drowned when I was about 8 years old and I remember it very clearly. I did exactly as is described -- I was face up, mouth just at the water level, arms flat, completely quiet. My parents -- who had two older children and spent their whole lives by the water -- spotted me and pulled me out.
The thing I remember most is that the overriding sensation was not panic, but calm -- a weird, fatalistic calm. I was gong to die: "oops". It really is an odd physiological response, and you have absolutely no control over it.