SNR of filter output
Using
$$\int\limits_{-\infty}^\infty|s(t)*h(t)|^2 \,\mathrm dt$$
requires that this integral exists – and that requires, unless an infinite amount of the energy of $s$ is in $\ker((*h))$ (i.e. gets mapped to 0) that $s$ is an energy signal, not only a power signal (as you're probably used to dealing with). (An energy signal is exactly a signal that is magnitude-integrable.)
For that, defining SNR as quotient of energies makes sense. Also, your $S(f), H(f)$ are then energy spectral densities, instead of power spectral densities.
Problem: WGN is not an energy signal, and it's by definition not in the kernel of any deterministic filter. Therefore,
$$\int\limits_{-\infty}^\infty|w(t)*h(t)|^2 \,\mathrm dt$$
simply doesn't exist and the formula doesn't &quot;work out&quot;, as it has a diverging denominator.
I can only guess where the formula came from, something like from the derivation of (average) power of a signal $\xi$ being $\lim\limits_{T\to\infty}\frac1T\int_{T/2}^{T/2} |\xi(t)|^2 \,\mathrm dt$, and then eager cancelation of the $\frac 1T$ from  denominator and numerator?