Hmm... so the D could be for 'D'egrees of freedom? Who knows... like someone said, it is marketing, and hence a lot could be lost in translation just for easy/popular optics.
What is the difference between dimensionality and degrees of freedom? The standard definition of degrees of freedom is "number of variables that make up the state of the system" and the definition of the dimensionality of the state vector is...
There is no difference. I think people just get hung up on the word 'dimension' in the sense of degrees of freedom in a spatial state vector (to reuse your terms). Perhaps these are people who learned linear algebra only in the context of spatial dimensions / geometry and have never had to apply the concepts to any other problem domain.