Norms like $\ell_p$, $p \ge 1$, or quasi-norms ($0<p< 1$) are all $1$-homogeneous: $\ell_p(\lambda x) = |\lambda|\ell_p( x)$. Which is not the case for the $\ell_0$ count measure, which is scale invariant: a signal and its scaled versions possess the same sparsity index, as they have the same quantity of zero values.
I have been thinking a lot about Hurley's paper in the past years. A sparsity index should be (close to) scale invariant. From an more error perspective, Gini is excellent, but some $\ell_p( x)/\ell_q( x)$ norm ratios are almost as good. From an optimization perspective, Gini remains difficult as a penalty, while some $\ell_p( x)/\ell_q( x)$ ratios of norms are, slowly, becoming tractable. An example, and useful references, are given in Audrey Repetti et al., 2015, IEEE Signal Processing Letters, Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed ℓ1/ℓ2 Regularization.
So I would use both to try to measure sparsity alone.
Having the same $\lambda $ with both Gini and $\ell_1( x)/\ell_2( x)$ is by itself an interesting result. Sparsity of the recovered signal is one thing. The statistics of the residuals is another thing. I do not know of any standard metric combining both of them. Indeed, they don't have the same physical units: any loss quasi-norm
$$\ell_p(A\hat{x}−y)$$
will be $1$-homogeneous with respect to $y$, while the ratio $$\ell_1( \hat{x})/\ell_2( \hat{x})$$ has "no unit".
So for a linear combination, one needs a coupling weight $\omega$:
$$\ell_p(A\hat{x}−y) + \omega\ell_1( \hat{x})/\ell_2( \hat{x})$$
and like in the regression model, how do you chose $\omega$?
I believe that a 2D metric should be better, but then you lose the possibility to have a natural order to compare two outcomes. In a similar context, I am also struggling to find a nice quantitative comparison of scaled-, delayed- and warped-signals.