The meaning of symmetry in field theories (probably a notational problem)

Question

I'm quite confused about the meaning of "symmetry" in the context of field theories. After reading many posts like 1-, 2-, 3-, 4- and 5-, I'm even overwhelmed.

My first approach would be the following: consider as an example, the Klein-Gordon field satisfying the equation of motion: \begin{equation} \partial_{\mu}\partial^{\mu}\phi(x) + m^2\phi(x) = 0\,. \end{equation} The dynamics of the field are described by this equation. This means that, after applying a transformation $\phi\rightarrow\phi'$ and $x\rightarrow x'$, for it to be a symmetry means that: \begin{equation} \partial'_{\mu}\partial'^{\mu}\phi'(x') + m^2\phi'(x') = 0\,, \end{equation} that is, there is a form-invariance of the equations of motion under the transformation. This ensures that the transformed fields describe the same dynamics, which is what we physically expect (in case I'm understanding things correctly).

Now, in the context of the variational principle, the equations of motion are obtained by considering a variation of the fields (and only of the fields, no variation of the coordinates) and extremizing the action: \begin{equation} S_{\Omega}[\phi] = \int_{\Omega}d^4x\,\mathcal{L}(\phi(x),\partial_{\mu}\phi(x),x)\,, \end{equation} with the corresponding boundary conditions for the variations of the fields. Without considering quasi-symmetries:

1. What does a symmetry imply: $S'_{\Omega'}[\phi']=S_{\Omega}[\phi]$ or $S_{\Omega'}[\phi']=S_{\Omega}[\phi]$? Here, the symbols mean: \begin{align} S'_{\Omega'}[\phi']&=\int_{\Omega'}d^4x'\,\mathcal{L}'(\phi'(x'),\partial'_{\mu}\phi'(x'),x')\\ S_{\Omega'}[\phi']&=\int_{\Omega'}d^4x'\,\mathcal{L}(\phi'(x'),\partial'_{\mu}\phi'(x'),x').\\ \end{align}

2. Does any of the previous options imply that: \begin{equation} \mathcal{L}'(\phi'(x'),\partial'_{\mu}\phi'(x'),x')=\mathcal{L}(\phi(x),\partial_{\mu}\phi(x),x)\,, \end{equation} or: \begin{equation} \mathcal{L}(\phi'(x'),\partial'_{\mu}\phi'(x'),x')=\mathcal{L}(\phi(x),\partial_{\mu}\phi(x),x)\,? \end{equation} And is there any relation between those equalities and the often encountered $\mathcal{L}'(\phi'(x'),\partial'_{\mu}\phi'(x'),x')=\mathcal{L}(\phi'(x'),\partial'_{\mu}\phi'(x'),x')$?

3. Are the transformations in my questions 1. and 2. different things? Like "symmetries of the action" and "symmetries of the Lagrangian"?

4. How is this all related to the original meaning of a symmetry that the equations of motion have a form-invariance under the transformation?