Fulton's accepts the premium and platinum package plans. That's why it comes up when you search DDP restaurants - even though the premium and platinum packages aren't the same as the Disney Dining Plans, because they have a dining component that is similar to the Disney Dining Plans, the website lumps all the plans together. If you were to click on the Dining Plan Information on the page you get when you make a reservation at Fulton's, you would see that it applies only to the Premium and Platinum Package Plans. A lot of guests, though, aren't going so far as to click a further link during the reservation process and are assuming dining plan meals all dining plans. Heck, most guests probably don't even know the premium and platinum package plans exist.
Yes, it's confusing - a lot of posters have been on here thinking that Fultons takes the dining plan - and I've been waiting for someone to come along and say they actually went to Fulton's thinking they could pay with the dining plan. The best way to deal with this is not through WDW-DINE (they have no idea what the website people are doing) but find some way to comment on the website. Most company websites have a "contact us" area for feedback.
It's not a question of "don't use the online system" but being aware of how they list dining plan/premium & platnium restaurants, which they didn't make easy to determine.
A better system for guests trying to determine what's on DDP is to use the printed DDP brochure (there is a link to one in the sticky threads of the Dining Plan board). If it's not on that brochure, it's not on DDP (make sure, though, that you are using an up to date brochure. A 2011 brochure should have a date in small print at the bottom of 121010; if it's dated any earlier, it won't be complete.)