And nowere does it define "Ranged Weapon" in any way that makes more sense.
You are saying "Range Weapon" is not the same thing as "Ranged Weapon" but then you are making up what a Ranged Weapon is without any sort of better definiton at all.
Your whole argument boils down to the words that are written in the PHB are not the rules because I don't like those words and think it should be this.
I was with you up until I saw there isn't a firm reliable hyperlink from the Sneak Attack definition to Range weapon.
If that's unreliable, then it makes as much sense to use table headers as it does to use the Range Weapon definition. After all, we already use table headers for some other parts of the rules. It's not like table headers have no meaning with these rules.
I can see the argument that the sneak attack rule is asking for a weapon CATEGORY (which is the table headers), where a weapon is either melee or ranged, and thrown doesn't make it a ranged weapon it just makes it a melee weapon which can make a ranged attack.
I can see the argument that the sneak attack rule is asking for a weapon PROPERTY, where a ranged weapon is one which includes either ammunition or thrown.
I think both of these views are completely rational.
I thought the confusion was broken by WOTC linking the sneak attack Ranged Weapon to the property rather than the category. But, that was just been demonstrated by @UngeheuerLichLegend to be unreliable because one version of the DNDBeyond rules has that link, and another doesn't, and I have no clue what that means or why there are two versions other than to confirm there are two versions.
So I guess my position now is I am with anyone who says they are unsure.