Tightslot,
my experience was with a US carrier in mid July - all the cabin crew asked pax was "are you a US resident / citizen" - my reply being 'no' I was given the white I-94 form + customs form. The mention of ESTA was non-existant.
Having gotten the ESTA authorisation I did wonder why I had been given the form but made the assumption that the cabin crew knew what they were doing re forms. I was aware that the green form had been abolished at certain airports but not 100% sure if my destination was one of them.