I know I already mentioned some of this and other places, but ‘present moment awareness’ for sati is an interpretation, whereas the context of sati as it is found in the suttas will help to guide interpretation.
When that context is Ānāpānasati and Satipaṭṭhāna, there are specific tasks sati is bringing ‘recollection’ to for contemplative work; with ānāpāna to establish an expansive calm with the breath, and with satipaṭṭhāna it is working with ātāpa and sampajāna to bring effort to examination of mind-states which arise at feeling.
“I say, beware of all enterprises that require new clothes, and not rather a new wearer of clothes.” – Henry David Thoreau, Walden, 1854
Secure your own mask before assisting others. – NORTHWEST AIRLINES (Pre-Flight Instruction)