I am sure my post was clear, which cannot be refuted by any deva, mara or sloppy scholar.
Mindfulness ('sati') means 'to remember'. The breathing as a meditation object cannot be 'remembered'. Instead the breathing is observed, which is called 'anupassi'.
The purpose of 'mindfulness' in Buddhism is 'to remember' to keep the mind in a state of right view & right intention (refer to MN 117) or 'to remember' to keep the mind free from covetousness & distress (refer to MN 118; MN 10; etc).
Therefore, to practise 'mindfulness of breathing' is an impossibility because mindfulness (sati) means to remember rather than to observe (anupassi).
Instead, the practise is 'mindfulness with breathing', which means to remember to keep the mind free in right view free from craving, as instructed in the four noble truths. When this is properly done, awareness/observation of breathing will naturally arise due to the quiet & silence of the mind.
Thus, mindfulness with breathing is remembering to keep the mind free from craving with awareness of breathing arising as the 'sign' of right mindfulness.