Stephen Wolfram provides an interesting description of dimension here. There are only a few citations but he says
The procedure we will follow is straightforward (cf. [1:p479][22]). For any point X in the graph define Vr(X) to be the number of points in the graph that can be reached by going at most graph distance r. This can be thought of as the volume of a ball of radius r in the graph centered at X.
That reference [22] is [22] C. Druţu and M. Kapovich (2018), Geometric Group Theory, Colloquium. in his references for that page.
This would suggest that I should look to page 479 of [22]. When I do, I see the end of a subsection on G-trees and the start of a section on An existence theorem for harmonic functions. The reference is not obvious.
Where did this notion of scale dimension come from?