So you've got a function with a known Fourier transform. If you want the Fourier transform of the derivative, you must use rule 1. If you want the Fourier transform of the integral, you must use rule 2. You can't use the rules interchangeably.

For the problem in the image you linked to, only rule 2 is applicable. The problem with using rule 1 is that h(t) is not uniquely determined given h'(t). I could define another h by taking your h and adding some constant. My h would be just as legitimate as yours, but would have a different Fourier transform.