"not skinny" includes both "fat" and "normal", though

In the pursuit of continuing to be charitable, I'll assume you're genuinely confused.

"Not skinny" does not mean "fat" necessarily, in the same way "not tall" does not necessarily mean "short", or "not cold" isn't the same as "hot". It's a continuum with a middle ground that's the assumed "default" so there are terms describing deviation from that middle. In common language, "not" preceding the term that describes that deviation tends to mean it falls within the default range, as if they meant it fell outside the range in the other direction they'd just use the term for that part of the range instead.

In this case, the assumed middle is the range of weights deemed "normal" so fat and skinny describe those body types that fall outside that range. Saying someone isn't skinny does not equate to saying they're fat since thay can be neither (and that's typically what's meant).