In my opinion, the reason why people are confused about this issue, is that DHS recently introduced a pre-departure APIS requirement whereby airlines are required to transmit to DHS passport information, including country of issuance, for all departing passengers.

Thus, an airline that takes passengers from NYC to Paris not only serves as an agent for the French Border Police (to whom they transmit the arrival APIS) but also serves as an agent for DHS (to whom they send the pre-departure APIS). My understanding is that this dual agency role is relatively new. Because DHS uses the pre-departure APIS for enforcement purposes, a reasonable person would conclude that the passport used for pre-departure APIS is the one you use to "exit the US."

The best solution here would be to convince airlines to allow passengers to have a departure APIS separate from the arrival APIS if they chose to do so.

Until then, my personal view is that one should always "exit the US" with a US passport.