I think Hawking, Ellis "The Large Scale Structure of Space-Time" 3.4 would be an interesting read for you.
From this book:
With Ricciscalar $R$, cosmological constant $\lambda$ and matter Lagrangian $L_m$
$$
I=\int_M (A (R - 2 \lambda ) + L_m)
$$
One might ask whether varying an action derived from some other
scalar combination of the metric and curvature tensors might not give
a reasonable alternative set of equations. However the curvature scalar
is the only such scalar linear in second derivatives of the metric tensor;
so only in this case can one transform away a surface integral and be
left with an equation involving only second derivatives of the metric.
If one tried any other scalar such as $R_{ab}R^{ab}$ or $R_{abcd}R^{abcd}$ one would
obtain an equation involving fourth derivatives of the metric tensor.
This would seem objectionable, as all other equations of physics are
first or second order. If the field equations were fourth order, it would
be necessary to specify not only the initial values of the metric and its
first derivatives, but also the second and third derivatives, in order to
determine the evolution of the metric.
Another nice read is Carroll "Spacetime and Geometry: An Introduction to General Relativity" 4.8. Alternative Theories. There is an shorter version online for free in Chapter 4 here
Of course, the metric $\eta_{\mu\nu}$ is not a unique solution for Einstein vacuum equations compatible with your given initial data. And yes, we can interpret the alternatives as arising from coordinate functions.
Let us take the simplest of such function: redefine time by introducing new 'time' variable $\tau$ through a relation $t=f(\tau)$ (spacial coordinates we will keep intact). The metric in new coordinates $(\tau,x,y,z)$ would be
$$
ds^2=(f'(\tau))^2 d\tau^2 - \delta_{ij}dx^i dx^j.
$$
It is, obviously, a different metric. And by choosing the function $f$ satisfying some simple conditions ($f(0)=0$, $f'(0)=1$, $f''(0)=0$) this metric will be compatible with your initial data.
But at the same time it is equally obvious that this metric still corresponds to the same space-time - the Minkowski space-time (at least locally).
Addition. To make a solution of Einstein equations unique one can use coordinate conditions (which are analogous to gauge fixing conditions in EM theory). These work as constraints on metric imposed in addition to Einstein equations.
Also, if you are interested in initial data - time evolution formulation of general relativity, I recommend looking at the ADM formalism.
Best Answer
The formal classification of solutions to Einstein field equations is the Penrose-Petrov-Pirani scheme. This classifies solutions according to eigen-Killing vectors of the Weyl tensor. These solutions range from the type D solutions for black holes to the type N solutions for gravity waves. In between there are type I II and III solutions for Robinson-Trautman spacetimes. These turn out to have a relationship to each other, where type D solutions are near source terms and type N solutions are far field. The other solutions are in between. This is a gravitational analogue of the near and far field solutions for Mazwell’s equations. A source on these solutions is in:
H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions of Einstein’s Field Equations, Cambridge: Cambridge University Press. (2003)