🚀 go-pugleaf

The boxcounting dimension (and algorithm) is an approximation to the rigorous
Hausdorff dimension, which is usually accepted as the "Fractal dimension". 
There are a multitude of other dimensions as well, such as information
dimension, simularity dimension, cluster dimension, etc.  These dimensions
usually, but not always, agree.  Note that boxcounting is a particularly
poor choice for estimation of the dimension of experimental data.  For a 
rigorous treatment of these subjects, see Falconer, "The Geometry of Fractal
Sets".  See Feder, "Fractals", for a good introduction, or the AMS collection
on fractals for a more rigorous but succinct treatment.

My own experience is that estimation of the Hurst exponent via range scaling
analysis is the best first approach.