A square is a kind of rectangle, nobody that says "draw a rectangle" expects you to draw a square, they expect a box with two sides longer than the other two but equal in length to each other adjoined at 90° angles. A square is a subcategory of rectangle. Its definition includes some features of a rectangle, but it also has other features that define it as clearly different from a rectangle.

Likewise a laptop is a subcategory of personal computer. A tablet such as a Microsoft Surface could also be a subcategory of personal computer. But if someone talks about their PC, you don't think of a laptop or Surface.

Its the same with consoles and handhelds. By technicality one could call a handheld a console in the same way a person could call a Surface a personal computer, but that would be the same as calling a hot dog a sandwich, or calling a submarine a boat.

Perhaps "home console" would be more descriptive, but since video game consoles were always understood to not be handheld devices, there is no need to subdivide the parent category. Just "consoles" and "handhelds" works fine.