Correction: it says unarmed strike or improvised weapon, both of which qualify as weapon attacks (based on interaction with other features).
The reason I bring up hits is because there are a few features, such as fire shield, that deal damage when an opponent hits the recipient with a melee attack. If a player shoved an enemy under the effects of fire shield, I suspect most DMs would say the player takes damage, even though the player didn't "hit" with a melee attack if we don't count shoves as hitting.
Consistency is key.