Sometimes it depends on the carriers also. I flew Virgin Atlantic, and at the Delhi airport the security folks for the airline (who were Indian) asked me for the AP. At that time I had 3 copies of the AP, and I told them that I need all 3, but they reminded me that one copy is for the carrier (the airline in this case). So I had to give my 3rd original (without the ADIT picture). They did however make a photcopy of the document that they had taken and gave me the photocopy.
Now it seems that the BCIS is only issuing 2 copies of the AP, so we can probably get away with giving a photocopy of the AP to the airline should they ask for it.