I may be off base here, but I think the distinction is between number of degrees of freedom (what you describe) and density of bits of information. Bekenstein's result showed that the information in a black hole is proportional to its surface area, not its volume. This seems unintuitive in the sense that if you, e.g., stuff more ram chips into a box, its information content seems to increase as a function of volume; the disparity lies in the fact that bits in ram are not as densely packed as in a black hole.
I think there is no distinction to be made. Consider encoding information in a 4-cube (hypercube) constrained by Maxwell's equations. (Let's make the space discrete so our heads don't explode.) You will find that you can only encode an amount of information proportional to the cube of the 4-cube's edge length, not the 4th power, because you are constrained by Maxwell's equations. If you further insist on keeping the information unchanged with respect to time, you'll be restricted to an amount proportional to the square of the 4-cube's edge length.
EDIT: misspelled Bekenstein