Post-errata, page 195 of the PHB contains the text:

On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier.

Moreover, page 196 says:

When attacking with a weapon, you add your ability modifier — the same modifier used for the attack roll — to the damage.

Presumably, an unarmed strike is a melee weapon attack, and thus by the rules on page 194 the ability modifier is the Strength modifier. Combining the two rules, this means that the final damage dealt by an unarmed strike is 1 + Str modifier + Str modifier, which seems a little... odd.

I can think of the following possibilities to explain what's going on here:

• Adding the Strength modifier twice is the right thing to do.




• Saying that the damage is 1 + the Strength modifier is just a reminder of the fact that you have to add the ability modifier, and thus you only add the modifier once when calculating damage. This is slightly supported by the fact that, pre-errata, unarmed strikes did just 1 damage. (If I'm interpreting it correctly, this also seems to be the conclusion reached in this question, though the reasoning there isn't exactly clear.)




• An unarmed strike is neither a melee weapon attack nor a ranged weapon attack, and thus the ability modifier on the attack roll is zero, and thus the Strength modifier has to be explicitly included in the damage definition in order for stronger characters to deal more damage.

So, which is it?

You only apply the strength modifier once because unarmed strikes don't involve a weapon

Unarmed strikes are considered melee weapon attacks, but don't involve a weapon

Unarmed strikes are melee weapon attacks despite not being made with a weapon.

The errata for the PHB states:

Weapons (p. 149). Unarmed strike doesn’t belong on the Weapons table.

This is even more clearly stated in the Sage Advice Compendium:

[...] an unarmed strike counts as a melee weapon attack, even though
the attacker’s body isn’t considered a weapon.

and

The game often makes exceptions to general rules, and this is an important exception: that unarmed strikes count as melee weapon attacks despite not being weapons.

Unarmed strikes only follow the rule for unarmed strikes

There are rules for how to do an attack with a weapon and rules for attacks from spells, but no rules for a melee weapon attack that has neither a weapon or from a spell. Unarmed strikes, being weird, fall into a crack in the rules.

Specifically, the second rule you mentioned does not apply because it is only for attacks with a weapon.

When attacking with a weapon [...]

This is the reason that unarmed strikes get their own rule:

On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier.

So you use this rule (and only this rule) to calculate the damage for an unarmed strike.