CleverNickName
Limit Break Dancing
In that case, if a player were to purchase a newer edition of the PHB and point it out to me at the table, then I would probably rule the reverse. The spirit of my ruling is pretty straightforward: if something (anything) is on that Melee Weapon Table, it's a "melee weapon." Otherwise, it isn't. And lots of stuff aren't: bites, claws, tail slaps, slams, and from the sound of it, unarmed strikes.Actually, it isn't - in later editions of the PHB. It was changed.
Now if a class feature, feat, or other overriding rule specifically says something else applies specifically (like with the monk's Martial Arts ability: "When you use the Attack action with an Unarmed Strike or a monk weapon...") it overrides.