In textbooks it's stated that one should renormalize the field strengths in addition to coarse graining and rescaling by a factor of for example $z$. For a scalar $\phi^4$ theory they choose $z$ to be such that the coefficient of the $|\nabla \phi |^2$ remains unchanged. Books reason that this choice of $z$ results in a RG flow with 2 relevant directions which is convenient to describe a ferromagnetic phase transition.
About the choice of $z$ I have seen three different opinions:
1-Textbooks claim that one can choose any other values for $z$ and he will get different coarse grained Hamiltonians with different critical behaviors.
2-In this question the accepted answer argues that different values of $z$ just give us different but "equivalent" RG flows.
3-For each theory the value of $z$ is uniquely determined.
Personally the second opinion is false as it's evident even in the Gaussian model that different $z$ s will give different RG flows. But the first statement also seems peculiar to me, since I think that one shouldn't renormalize the field variables if he attempts to find the effective Hamiltonian of the system when the system is observed over larger length scales (if you view an image from a larger distance, your eyes don't enhance the intensity of light which comes from the image). Another problem with the first statement is that why should two systems with exactly the same statistical field theories have different critical behaviors?
And if the third statement is true, what're the criteria according to which we should define $z$ ? Because apparently people define $z$ in different contexts with different reasons. For example in treating the non-linear sigma model it's not important that how many relevant parameters the coarse grained Hamiltonian will have.