It's a combination of all of the above, actually.
Though Zero inflicted a similar wound to one side of Sigma's face, he also tore of the entirety of Sigma's skin on the other. Therefore it would only account for one of the two scars. Furthermore, Sigma had gotten his whole face repaired after the incident. However, his loss against Zero definitely was a crushing blow to his ego and is clearly remembered by him.
It is after X's attack that Sigma left the scars he was dealt with, sort of like a symbolic gesture of how X's influenced Sigma. But in that same way, it is also a symbolic gesture toward Zero, who once inflicted a similar wound. Indeed, as we all know, it were both X and Zero that were unwillingly responsible for Sigma's change of allegiance.
Then, with his subsequent revivals, Sigma's scars never went away; he keeps on re-adding them, reminding X and Zero of their connection.
What the manga showed doesn't apply to the games, though. This is because Day of Sigma presents a completely different scenario of Sigma revealing his rebellion.