I would would have said it naturally converges on whatever is appropriate to be aware of at the time. If at the time one is practising anapanasati then yes, if practising a different technique then maybe not, if eating lunch or walking or doing any number of other activities then awareness should converge of the activity at hand not on the in & out breathing specifically.
Placing ones awareness on breathing in & out is a very good way of developing the minds capacity to be aware, probably the best way to start for beginners, but it's not the only way.
To me, "watching the breath" is not Anapanasati, it is "watching the breath."