@TobiasFünke Don’t take me wrong, I’m not advocating for these current times were everybody wants to dismiss everything as just “an opinion”. I believe there are different kinds of opinions. Some are informed other not; some are pure speculative other not; some try to adjust to data other not; some come after though analyses of the problem other not. But we should avoid the abuse of the label “opinion” to dismiss current physics discussion.

@TobiasFünke What’s the difference of an informed opinion and a theory? The fact that we are constantly outdating fantastic theories of one’s time, just points that those great theories were no other than useful opinions that satisficed the evidence and experiments of their time. You adjust your arguments on the existing and foreseen data, but you still have to include an argument on the data, just as much a lawyer do in a court of law.

It can be informative to show that the last equation is not $E=(\gamma m)c^2$ but $E=m(\gamma c)c$ instead. Another linked misconception leading to "relativistic mass" is that momentum is $p=(\gamma m)v$. However, it actually is $p=m(\gamma v)$. $\gamma$ in these equations is transforming the speed, not the mass!

@KP99 I edited the question to address your concerns. I guess the shortcut I made by not mentioning that $f_{\mu \nu}(\mathbf{x})=\Lambda (\mathbf{x}) g_{\mu \nu}$ was not very clear there. More importantly, a mentioned $\beta g_{\mu \nu}$ as the integration constant only before, when in fact I meant $G_{\mu \nu}+\beta g_{\mu \nu}$. My mistake. Thanks for your comment :)

@KP99 Me too. I’m just showing to the OP what it means by replacing $\Lambda \in \Bbb{R}$ a constant, into $\Lambda=\Lambda (\mathbf{x})$ a function of the coordinates (time and space) in EFE. I only introduced $f_{\mu \nu}=\Lambda (\mathbf{x})g_{\mu \nu}$ to get into a more general form of that substitution, and no matter the solution $f$, you get from $(3)$, it is always possible to add $G_{\mu \nu}+\beta g_{\mu \nu}$ into it, making $(4)$ the most general form of the EFE.