Inclined planes form a column. Unguided inclined planes: knives, axes. Guided: planes, scissors. Inclined planes wrapped around cylinders: drills, screws.

Blunt objects form a column: Hammers, presses, brakes.

I agree with your categorizations involving tools that rely on heat, but I guess the problem with any such taxonomy is the next person comes along and goes "well I would do this differently, and that..."