So when he said that the distinction between a ranged weapon and melee weapon is under the headings, he was correct. The property is not what makes something melee or ranged.
It absolutely is. It is defined on page 214.
There's no reason to assume that just because the finesse property allows sneak attack, that the range(not ranged) property does as well. A melee weapon with the range property is still a melee weapon, not a ranged weapon.
Range Weapon is both defined and linked from the sneak attack section.