The distinction between X-Rays and gamma rays is not the way how they are produced but the energy level.
As others have pointed out, this is false. Here's a simple guide to the complex language of electromagnetic radiation:
1) If it was produced by an atomic process it's an x-ray, no matter what the energy.
2) If it was produced by a nuclear process, it's a gamma-ray, no matter what the energy
3) If the source is neither atomic nor nuclear, or unknown, it's field-dependent and circumstance dependent. I tend to think of bremstrahlung as gamma radiation unless I'm talking about x-ray sources for imaging or medical treatment. This is a purely cultural difference, with the terms "x-ray" and "gamma ray" being understood as interchangeable by practitioners, but with one or the other being preferred depending on context. Annihilation radiation is called gamma or x-ray depending on the field as well.
With regard to the EM radiation from storms, there are multiple possible origins. It's pretty easy to create neutrons from high-energy plasmas, as in the Farnsworth Fusor. Subsequent capture of those neutrons on nuclei will produce "true" gamma rays. On the other hand, various purely EM processes could be producing x-rays as well. So the EM radiation from storms could well be a mix of both nuclear and atomic processes. Call 'em gammas or x-rays, and don't make a big deal of it.