That is incorrect. OP, you are correct, as long as you check in by the last day of a free dining promo, you can have the FD discount applied to your entire stay, up to 14 nights.
Free Dining Plans and other "full package" promotions will cover your entire stay (up to the maximum, in this case, 14 nights) so long as you check-in before the deadline.
Room only promotions (e.g. discounts) typically go night by night, so if you check in on a 25% room discount on the last day, you'll only get discounted for that night (this is exactly what made the free dining plan a better option for me last year over the 30% discount, even as a solo, since my room would have been discounted for 1 night out of 10 )