You don’t get to define words
I don’t know where you got your definition of “socialism”, and in any case it doesn’t matter, because no one gets to define words. Not you, not even dictionaries … as any lexicographer will tell you. Words mean what people think they mean, and dictionaries only report it after the fact. (Same problem as people saying “Islam is a religion of peace.” Maybe for the person who says it, but for a Daesh sex-slave owner hell-bent on taking over the world, it’s a religion of violence.)
So if people say Bernie is a socialist, he is … for them. For them, that’s what the word means, end of story.
It’s interesting to see you trying to assert a claim in connection with this word because you are practicing exactly the methodology of authoritarianism, of which socialism is one manifestation.
According to me, anyway! :)
Peace, love and bunny-rabbits.