Two perspectives on families of m-ary trees. The second coincides with your formalism and provides two stages of generalization (and hints at a third). The first is related to comments by others.
First perspective on types of n-ary trees, factorials and generalizations:
$F_t(x) = (1-tx)^{-1/t} - 1$
is the e.g.f. for enumerating plane $m$-ary trees when $t=m-1$. OEIS A094638 provides some examples when $t = \pm 1,\pm2,\pm3$ in my Dec. 15, 2007, comments. The e.g.f. is of importance for $t$ any real number.
For $t = 3$, the e.g.f. is
$F_3(x) = (1-3x)^{-1/3} - 1 = x + 4 x^2/2! + 28 x^3/3! + 280 x^4/4! + \cdots$,
generating the sequence A007559, the right triple factorials, enumerating the number of increasing quaternary trees on n vertices.
For $t = -3$, the e.g.f. is
$F_{-3}(x) = (1+3x)^{1/3}-1 = x - 2x^2/2! + 10 x^3/3! - 80 x^4/4! + \cdots$,
generating the signed sequence A008544, the left triple factorials, enumerating increasing plane (a.k.a. ordered) trees with n vertices (one of them a root labeled 1) where each vertex with outdegree r >= 0 comes in r+1 types (like an (r+1)-ary vertex).
Note the relation to multiplicative inversion:
$1+F_t(x) = 1/(1+F_{-t}(-x))$.
(I called this inversion, A133314, the list partition transform, in the A094638 for historical reasons in my journey of discovery of its significance.)
The compositional inverse (CI) plays a role also. The CI in $x$ of $G(x,t)$ about the origin $x=0$ of $F(x,t) = F_t(x)$ is
$G(x,t)= [1-(1+x)^{-t}]/t$,
so the infinitesimal generator / Lie vector for generating $F(x,t)$ is
$g(x,t)\partial_x = \frac{1}{G'(x,t)}\partial_x = (1+x)^{t+1}\partial_x$;
that is,
$\exp[x g(z,t)\partial_z] z |_{z=0}= F(x,t)$.
The Cayley analytic trees associated with the iterated operation
$(g(z)\partial_z)^n z |_{z=0}$
for $t=\pm3$ are of the types described above (see. e.g., my "Mathemagical Forests" reffed in A145271 along with the Bergeron et al. ref "Varieties of Trees" and my "Addendum to Mathemagical Forests" in A094638). (This is all related to the formalism of pre-Lie algebras.)
My formulas in OEIS A094638, as pointed out therein and above, involve both compositional inversion and multiplicative inversion (A133314) and so are naturally related to Koszul duality as noted in the MO-Q "Inversion, Koszul duality, combinatorics and geometry". In fact, as I remarked in the formula section of A094638 and repeat above: With
$F(x,t) = (1-tx)^{-1/t} - 1$
an e.g.f. for the row polynomials $P(n,t)$ of A094638 with $P(0,t)=0$,
$G(x,t)= [1-(1+x)^{-t}]/t$
is the CI in $x$ about $x=0$. The case for $t=3$ is used in the proof of Corollary 4.2. of "Associator dependent algebras and Koszul duality" by Bremner and Dotsenko.
Second perspective on types of n-ary trees, Euler-Fuss-Catalan numbers and generalizations:
My comments on the relation between $[A^{(m)}]$ for $m \geq 1$ and the Fuss-Catalan numbers, generated by compositional inversion of $f(x) = x \pm x^{m+1}$ about $x=0$ (see, e.g., A001764), in my answer / extension to the MO-Q "Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution" (IDI) presents another perspective since $m$ is extended there to any integer. This is a generalization of the formalism of Novelli and Thibon in "Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions".
I have proved, in notes I'll soon post on my blog, that the partition polynomials presented in IDI naturally reduce to variants of the row polynomials of the triangles compiled in the table on p. 23 of "On the inversion of Riordan arrays" by Paul Barry and also discussed by N & T. Note the first columns of the coefficient triangles in the right column of the table for $m=-1,2,3,4$ contain variants of the Fuss-Catalan sequences, A000108, A000108 (again), A001764, and A002293. This is because the full right triangles correspond to reductions to the non-vanishing partition polynomials of the compositional inversions of
$O^{(p)}(x) = x + c_1x^{p+1} + c_2x^{2p+1}+c_3x^{3p+1}+\cdots$
with $c_1 = \alpha$ and other $c_k =1$ (or equivalent reductions), which, in turn, reduces to
$O_{red}^{(p)}(x) = x + \alpha x^{p+1}$
for $c_k =0$ for $k>1$ giving the first columns of the triangles as coefficients (mod signs and index shifts) of the non-zero coefficients of $(O_{red}^{(p)}(x))^{(-1)}$, the generating fcts. for the aerated Fuss-Catalan sequences for $m \geq 2$.
The right column of Barry's table contains natural reductions (mod signs and reversals of order of coefficients) of $[A^{(-2)}]$ for $m=-1$ to A286784, $[A^{(-1)}]$ for $m=0$ to A090181 / A001263, $[A^{(0)}]$ for $m=1$ to A007318, $[A^{(1)}]$ for $m=2$ to A126216 / A033282 / A086810, $[A^{(2)}]$ for $m=3$ to A243662 / A102537, and $[A^{(3)}]$ for $m=4$ to A24366. In this light, the compositional inversion identity (CII) in item 11 of "Guises of the noncrossing partitions (NCPs)" appears as a generalization of your algebraic characterization of the o.g.f. of types of n-ary trees.
On Fuss-Catalan numbers and relations among their generating functions, see Wikipedia and also pages 33 and 34 of "Connecting Scalar Amplitudes using The Positive Tropical Grassmannian" by Cachazo and Umbert.
I've been a little too busy putting my notes with relevant proofs into pdfs to look at the explicit coefficients of $[A^{(m)}]$ for $m < -2$. However, Barry gives a formula in his table from which I computed the unsigned first columns of the triangles:
$a_n = \frac{1}{n+1} \binom{|m+1|(n+1)-1+n}{n}$
for $m=-3$ down, giving A006013, A006632, and A118971, which led to p. 19 of the ref "Some relatives of the Catalan sequence" by Liszewska and Młotkowski in A118971, through which this list continues as A130564, A130565, A234466, A234513, A234573, and A235340, consistent with spot checks by Barry's formula.
L & M start with the equation
$B_p(z) = 1 + zB_p(z)^p$
satisfied by
$B_p(z) = \sum_{n \ge 0}\frac{1}{n+1} \binom{np+1}{n} z^n$
and state that Lambert found
$B_p(z)^r = \sum_{n \ge 0}\frac{1}{np+r} \binom{np+r}{n} z^n$
valid even for $p$ and $r$ real. Then
$B_{-m}(z)^1 = \sum_{n \ge 0}\frac{1}{-mn+1} \binom{-mn+1}{n} z^n$
and using $\binom{-q}{n} = (-1)^n \binom{-q-1+n}{n}$,
$B_{-m}(z)^1 = \sum_{n \ge 0}(-1)^{n} \frac{1}{-mn+1} \binom{mn-1-1+n}{n} z^n$,
which yields the same sequence of absolute numbers as the formula I have from Barry.
Edit Mar 6, 2023, A combinatorial interpretation: (START) The coefficient $\frac{1}{-mn+1} \binom{(m+1)n-2}{n}$ of $B_{-m}(z)^1$ is presented on page 9 of "Refined Lattice Path Enumeration and Combinatorial Reciprocity" by Henri Mühle and Eleni Tzanaki. They denote this as $Cat^{+}(m,n)$, the positive Fuss-Catalan number, and state it is the sum of the (reverse) positive Fuss–Narayana numbers. They claim,"Surprisingly little is known about combinatorial interpretations of the (reverse) positive Fuß–Narayana numbers. They count the h-vector of the positive m-cluster complex [4, Corollary 5.5]. In reverse order, they count the number of bounded dominant regions in the type-A Catalan arrangement according to the number of their non-separating walls of type xi − xj = m; [4, Corollary 4.4]."
On p. 28, in the section 4.3. Ehrhart reciprocity, M & T define the usual Fuss-Catalan numbers as $Cat(m,n) = \frac{1}{mn+1} \binom{(m+1)n}{n}$ and claim an instance of the Ehrhart reciprocity manifests in the equality
$Cat(-m,n) = (-1)^{n -1}Cat^{+}(m-1,n),$
which I've demonstrated above. They continue with "The key fact behind this reciprocity is that the dominant regions of the Catalan arrangement, which are enumerated by $Cat(m, n)$, are in bijection with the integer points of a certain simplex (see [2, Section 4]). An analogous bijection holds for the bounded dominant regions, which are enumerated by $Cat^{+}(m, n)$ 4."
Some more on this, on generalizations to Coxeter groups other than $A_{n}$, and on the connections between the noncrossing partitions / refined Narayana and associahedra partition polynomials can be found in Drew Armstrong's Ph.D. thesis "Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups". (For an example of the relation between $[A^{(2)}]$ and $[N^{(2)}]$ presented in IDI, but in Armstrong's terminology , see p. 234.)
(END)
The Fuss-Catalan numbers can be defined via compositional inversion and so can the positive Fuss-Catalan numbers (maybe negative would be a better qualifier from my analyses).
Letting $FC_p(z) = zB_p(z)$, the defining equation becomes
$FC_p(z) = z + (FC_p(z))^p$.
Then the compositional inverse satisfies
$FC_p^{(-1)}(z) = z - z^p,$
and, for $p = 2,3,4,5$, this is the CI for the Fuss-Catalan sequences (aerated and signed) noted above--A000108, A001764, A002293, ... .
For $p = 0, -1,-2,-3,...$, this is a Laurent series for which Example III of the MO-Q "Infinite dimensional involutions ..." applies to obtain the CI with $a_0 =1$, $a_{1-p} = -1$, and otherwise $a_k=0$. The partition polynomials of this example are those of A355201 with a formula for their numerical coefficients and a combinatorial model for the formula in my pdf "One Matrix to Rule Them All ..." (link in the OEIS entry). (See the comments below for an example.)
I would search for other overarching combinatorial models in comments and refs in the pertinent OEIS entries, e.g., the ref "Multivariate Fuss-Catalan numbers" by Aval in A235340. I need some time to confirm the relationship to that article, and there's the ref "Coding of ordered trees" in A130564, to which I have no access.
As an overarching combinatorial construct for encompassing all the integers $m$ for the sets of partition polynomials $[A^{(m)}]$ (and $[N^{(m)}]$), I'd like to see a comprehensive theory of their relation to Feynman diagrams / Green functions in QFT since this is a common thread in discussions of the related triangles down to $m=-2$. Balduf, Yeats, Kreimer, their collaborators, and various other researchers have investigated this for $m >1$ as related to Schwinger-Dyson equations and Hopf algebras. N & T give also a more refined situation for which the indeterminates in the multivariate partition polynomials (at least for $m$ positive) are noncommutative.