Yeah, I think he's using the definition of dimension as per mathematics and physics...1-dimensional, 2-dimensional, 3-dimensional, etc. Hyperspace being the 4th or 5th or whatever-th dimensional reference frame.
We're using dimensions like the typical squishy sci-fi writer does, more like a parallel reality/universe, the sci-fi- version of a fantasy setting's planes. The way I'm seeing it, Hyperspace is kind of like the Ethereal Plane in the D&D cosmology- concurrent with meatspace and overlapping it everywhere, and one can exist partway between them or transit from one to the other easily and anywhere. If turning Ethereal made you move faster instead of slower (to replicate the 'hyperspace is more densely packed' effect), it'd be a perfect analogy.
The Warp would be more like the Astral Plane - it can be reached from anywhere in the real world, and touches everywhere in the real world, but no point in the Astral can be correlated to any actual point in meatspace, parallel but non-concurrent.