The naming convention used is a bit confusing. Do the terms melee attack and melee weapon attack mean the same thing if unarmed strikes are considered the latter?
This is what I've gathered on the topic, using the Monster Manual as my example guide...
Attacks can be broken into two categories: range (Melee or Ranged) and Type (Weapon or Spell). You can mix and match the two to create different types of attacks.
Melee+Weapon: longsword, dagger, bite, claw, unarmed strike. Uses Strength unless a rule (finesse, monk) says otherwise.
Ranged+Weapon: longbow, crossbow, thrown dagger, blight needles, tail spines. Uses Dexterity unless a rule (thrown property) says otherwise.
Spell+Melee: Flameblade, shocking grasp, intoxicating touch, life drain. Ability score varies based on Caster/Monster (but typically Int/Wis/Cha)
Spell+Ranged: scorching ray, ray of frost, demon/devil hurl fire, empyrian bolt. Ability score varies based on Caster/Monster.
If the rules refer to a "Melee weapon attack" They are only referring to attacks made with a natural (claw/strike) or manufactured weapon (sword, axe). If they refer to a "melee attack", it can be either a Weapon or Spell attack made in melee. If they refer to a Weapon Attack, its a Melee or Ranged natural/manufactured weapon attack. But if they refer to a Weapon (no attack) they are referring only to the list of manufactured weapons (swords, axes) but not natural weapons.
Yeah, the terminology could've been clearer.
This also ties into Damage Resistance; the PHB errata added a touch of MM errata that should read "bludgeoning, piercing, and slashing from nonmagical weapons ATTACKS", otherwise a longsword would be resisted but a claw, bite, or unarmed strike would not. Crawford clarified this somewhere.
EDIT: Fixed Important Typo.