Your interpretation is incorrect. The only definition of ranged and melee weapons is the table.
I just quoted you a definition; the table is an illustration.
In this case you assume that ""A melee weapon is used to attack a target within 5 feet of you" is equivalent to saying " A weapon used to attack a target within 5 feet of you is a melee weapon". That is not what the sentence says. The sentence says that melee weapons are in the set of things that can be used to attack a target within 5 feet of you. It does not say that the set of things that can be used to attack a target within 5 feet of you is the set of melee weapons. A more clarifying example of this fallacy is recognizing that
"A dog is a mammal" does not imply that "A mammal is a dog."
Tools are not animals and are not subject to unary classification. What is the difference between a knife and a dagger? a dagger and a short sword? a short sword and a long sword? Between a knife intended to be used in a kitchen (a tool) and a knife intended to be used in a fight (a weapon - or is that a tool also?)? If I attack someone with a kitchen knife, I will be charged with "assault with deadly weapon", so which is it?
If I fill the bed of my busted pick-up truck with dirt and plant it full of gardenias is it a truck or a garden? Do I still own a truck?
Things can belong to more than one category. My interpretation of the weapons definition is exactly that - a weapon is classified according to how it is used. The thrown property is still relevant and meaningful, as it determines which attribute may be used to modify attack rolls and damage.
As to
Cernor's concern that thrown weapon users would be "broken" because they could get as many attacks as other so-called "broken" builds: The thrown weapon user is still limited by a) how fast s/he can
draw new weapons and b) is still hampered by
limited range. A Sharpshooter throwing handaxes
can throw 3 if they start the turn with one in each hand.
Furthermore, they are even more limited because they
end the turn empty-handed, and so c)
can not repeat the trick next round and d) cannot use their reaction to
take an AoO with a weapon. Also, e) using an
action surge only adds 1 more attack, because of the limit on drawing weapons. Finally, f) magic weapons affect
1 attack per fight - 'cause you just threw it away.
Crossbow Expert/Sharpshooter is far, far more dangerous.