I mean, there isn't any rule differentiating ballistic and impact damages? I remember that in the old 2nd edition this rule existed, but I couldn't find something similar in the 5th and I would like to confirm that the armor protection is the same for all kinds of damage.
The removal of separate ballistic and impact armor ratings was a change from SR4 (and earlier editions) to SR5. There's not really any specific rules passage to quote, because it's a wholesale change between editions.
That said, the two attack examples on p. 174 of the SR5 core book are decent evidence of the lack of difference. One is ranged combat and the other is melee combat. They reference different armor pieces (jacket vs. lined coat), but neither example cites any need to do anything different with the related stats (Armor rating & AP rating) just because the attack is melee or ranged.
As for unarmed versus weapon attacks, that's just a function of damage type - some are Physical, others are Stun, some have elemental effects (like shock gloves). The rules use the AP value to control how well (or poorly) things penetrate armor, but again, the source of the attack doesn't impose any blanket differences. In effect "unarmed" is not a property of the attack. In the absence of specific text, it's just a description. It has no baseline mechanical significance, unless something adds it. There are some abilities (mostly Adept powers) that don't function with weapon, for instance.
For the record, in prior editions an attack being made without weapons didn't have any mechanical significance either.