Personality is more than just interaction in that it involves how you fundamentally process and react to information.
A person with a genial personality might get his foot stepped on in line at the movie theater. He processes the fact that his foot was stepped on, decides that it was obviously accidental, and does not react, except perhaps to politely request that the person not step on his foot.
Now take someone with a less genial personality, perhaps a burly redneck with poor impulse control and a tendency towards violent behaviour. He processes the fact that his foot was stepped on, decides that the person who stepped on his foot is homosexual, because a decent, God-fearing Christian heterosexual would never step on someone's foot like that. He then proceeds to beat the living crap out of the poor schmuck who stepped on his foot.