Non-uniqueness of the Order Parameter and its Critical Exponent
I think the conventional wisdom is correct, the choice of order parameter does not matter (as long as the symmetries are preserved, and we redefine conjugate variables).
I think the main mistake in the post is that you assume that if $M$ scales as $M\sim t^\beta$ then $M^a$ scales as $M^a\sim t^{a\beta}$. But this is not the case, $M$ is a stochastic variable, and the scaling behavior is $\langle M\rangle \sim t^\beta$. There is no simple relation between the scaling of $\langle M\rangle$ and $\langle M^a\rangle$. Also note that using $M^2$ is not a good example, because $M^2$ has different symmetries (and is indeed not an order parameter in the Ising case).
The way to investigate this is to start from the Landau-Ginzburg functional $$ {\cal F} = \int d^3x \left(\gamma(\nabla M)^2 +\alpha M^2 +\beta M^4 + \ldots +Mh\right) $$ and perform a change of variables. As long as the new variable has the same symmetries as the old variable the form of the LG functional is not changed, and the universal predictions remain the same.
The viewpoint of classical statistical mechanics is indeed that the order parameter is non-unique. For example, in his excellent Lectures On Phase Transitions And The Renormalization Group, Goldenfeld writes
The order parameter for a given system is not unique; any thermodynamic variable that is zero in the un-ordered phase and non-zero in an adjacent (on the phase diagram), usually ordered phase, is a possible choice for an order parameter. Trivially, we could perfectly well choose $M^3$ as the order parameter in a ferromagnet.
Unfortunately, I don't think he really dwells on the consequences of such a choice. As you've noted in the question, such a functional redefinition of the order parameter generically leads to changed critical exponents.
In the Landau-Ginzburg formalism, the Landau functional is given in powers of the order parameter and its gradients. I believe this should restrict the allowable functional redefinitions to analytical functions. It's helpful to further restrict this definition by requiring that the order parameter be a function of observables. This is still not unique, and is one reason for the debate about what the right order parameter is (if any) for e.g. the Mott transition.
Next we can consider the coupling to some external parameter (e.g. a magnetic field, which would give the term $\mathbf{M}\cdot\mathbf{H}$). The first derivative of the Landau functional with respect to this external field is precisely the order parameter1. This is kind of circular, because we could also redefine the field. However, the model is obviously most useful when the external parameter is chosen as a physical parameter or field. And if the coupling between the field and observables is known, this fact can be used to fix2 the nature of the $\mathbf{M}$ in the $\mathbf{M}\cdot\mathbf{H}$ term. In practice, this typically leads to an order parameter that's a linear observable, as Nortbert Shuch pointed out in a comment.
So, yes, when people talk about the order parameter or the critical exponent for a certain transition, they do tend to assume that everyone's talking about the same quantity. However, it should also generally be a physically motivated order parameter that couples to an appropriate field.
1: The first derivative can also include higher-order terms in the order parameter, but we can still isolate the lowest-order term.
2: Keep in mind that the Landau functional is, at least in principle, possible to derive as a mean-field theory of the underlying Hamiltonian.