$q$-analogs of total positivity

This is a wonderful question but unfortunately I don't think that there is a definite answer in the literature just yet. Let's look at two somewhat recent lines of research in this direction:

1) A. Sokal outlines a project of understanding what he calls coefficientwise total positivity in this research proposal (see theme #2, and also the referenced talk "Coefficientwise total positivity (via continued fractions) for some Hankel matrices of combinatorial polynomials"). His definition is identical to yours, but the only results they have are focused on Hankel type matrices, and only specific ones with various combinatorially meaningful coefficients. At the end of the talk there are a few open questions about establishing coefficientwise total positivity for some combinatorial sequences and it is mentioned that empirical observations show something stronger: not only are all minors coefficientwise positive, but their coefficients are also log-concave. I personally already take this as a hint that just asking for coefficientwise positivity might not be restrictive enough to "capture combinatorics". This brings us to the next proposal.

2) One first thought is that there is no reason to stop at just polynomials, we might as well extend to power series $\mathbb R[[q]]$ or formal Laurent series $\mathbb R((q))$. If we just focus on the simplest instance of defining the object $GL_1(\mathbb R[[q]])_{\geq 0}$, there is a classical definition of a totally positive function $a_0+a_1q+a_2q^2+\cdots$ (also called Polya frequency sequence/function) as one which satisfies the property of its associated infinite Toeplitz matrix, $\big(a_{j-i}\big)_{i,j\in \mathbb Z}$ being totally nonnegative (confusing teminology, I know). There is a rich theory around this notion, notable examples including the Edrei-Thoma classification of all such series, and the connection to the asymptotic representation of the symmetric groups.

The way I think of this is to think of the entire ring $\mathbb R[[q]]$ (or $\mathbb R((q))$) as a ring of infinite Toeplitz matrices, where $1$ corresponds to the identity and $q$ corresponds to the shift matrix. This way an arbitrary element corresponds to a Toeplitz matrix (with appropriate entries that are south-west enough being set to zero) and positivity of an element under the classical definition is the total nonnegativity of the corresponding Toeplitz matrix.

This is what allows Lam and Pylyavskyy to think of an element of $GL_n(\mathbb R((q)))$ as a matrix of matrices (aka the obvious block-Toeplitz matrix) and defining the totally nonnegative part as the elements whose associated block-Toeplitz matrix is totally nonnegative (there is also a slightly technical definition for total positivity instead of nonnegativity). The theory is spelled out in the papers "Total positivity in loop groups" I and II. The nice thing is that this q-analog of total positivity seems very promising. It shows structural similarity to the classical case: there is an analogous relation to cylindrical networks, an analogous generalization of the Edrei-Thoma theorem, the relation to Chevalley generators etc. However the generalization to other types is missing (in the first paper it is listed as "in preparation"). The authors mention that this line of thought might lead to "asymptotic cluster algebras", the same way that the classical picture of total positivity lead to the discovery of cluster algebras.