gresmi-
Last year the offer had the same wording, and I received 2X miles for all inter-island flights ACTUALLY FLOWN during the stated inclusive dates. Again be certain to give HA your Onepass number, and not a HA number. They do not need to be code-shared or other special type of flights. Any HA inter-island flight should get the miles.
Please note, it will take 1-3 mo to get credit and for a couple of flights I had to send copies of the HA boarding passes (real difficult to do using HI residents coupons) to get the miles. Overall, still it is a good deal.