There is no higher rate. Dolphin is a Sheraton, and 2 adults and 2 children are considered regular occupancy. The OP is not clear regarding who was asking for the higher rate. I know some booking sites can be glitchy, and will charge additional people as additional adults, even if the extra people are kids who don't trigger extra fees from the hotel.
DH stays at Marriott brand hotels all the time, and often books only under his name. We all show up, my name is added at check in, and they note number of kids.
When DH booked the Swan, he didn't realize the importance of having me and the boys on the reservation for Disney purposes (because I'm the Disney planner in the family). We got it sorted out over the phone, and there was no added fee.