Edit: *Hilbert Derived the EFE before Einstein, but using the EH Action as a postulate. Einstein took the EFE as a postulate.*

I think the matter is only the formulation of General Relativity; the postulates made.

**Hilbert's** postulate was elegantly simple. It was only that

$$\mathcal L_G=\lambda R$$

I.e. that the Lagrangian Density of Gravity was proportional to the Ricci Scalar. From then on, he applied the Hamilton's principle to this's equation and found that the field equations for gravity are $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ And then, he showed how this is a better theory of gravity etc. etc.

**Einstein's** postulate was not so elegant or simple, in my opinion. However, many hold that it was simpler. His postulate was the field equation itself! I.e that

$$G_{\mu\nu}=\kappa T_{\mu\nu}$$

The reason why people hold this to be very elegant is that $\nabla^\mu G_{\mu\nu}=0$ and $\nabla^\mu T_{\mu\nu}=0$.

So, it is no different than whether Einstein or Minkowski should get the credit. Minkowski did it the elegant way.

Similarly, Hilbert did it the elegant way.

**Edit:**

**Note:** That Einstein's postulate was the EFE has its reference in Einstein's original GR paper "On the Foundations of the General Theory of Relativity". As for Hilbert's postulate being the EH Action, I think it was from some book... Can't remember.

P.S. The fact that Einstein's postulate itself was the EFE means that he first wrote down the EFE. The fact that Hilbert's postulate itself was the EH Action means that he first wrote down the EH Action, (that is, 5 days before Einstein wrote down the EFE).

P.P.S. By "wrote down", I mean "published", who knows wheter Einstein/Hilbert actually wrote down the action/FE, or forced someone else to write it for them? Who knows...

P.P.P.S. Another important point of this answer is that there is no controversy, since it is just the formulation that matters, like Einstein&Minkowski, Newton&Leibniz, Heisenberg&Schrodinger&Feynman&somerandompersonwhodiscoveredthevariationalformulation&..., etc. Because the OP seems to think there's a big controversy... But there's not, at least there shouldn't be.