Fluctuation dissipation theorem : how to identify the response variable and the force in general?

How can I know which quantity should represent the input (generalized force) and output (response variable) so that in fits in the usual framework in which it is applied in physics.

In the standard setup of linear response theory, the Hamiltonian contains the product of the input $F$ and the output $x$, $$H_{\text{int}} \supset F(t) x.$$ Examples of pairs of this form include force and position, pressure and volume, and external magnetic field and magnetization. This is similar to the definition of conjugate variables in thermodynamics, since differentially we have $dU = F \, dx$.

Indeed up to my understanding, all the theorem in linear response theory are "simply" mathematical derivations. It is when we do physics that we say "this represents dissipation". Thus I expect in principle that we can take any variable as force and any variable as response

I wouldn't agree with that at all. The trivial parts of linear response theory are indeed independent of what you choose to be the input and output, since they follow from the symmetries alone. But statements such as the fluctuation dissipation theorem are proven starting from the assumption I made above. Of course, you cannot say anything whatsoever about energy dissipation unless you assume something about the Hamiltonian.