Linking an actual poll - for instance
https://news.gallup.com/poll/694835/image-capitalism-slips.aspx
Just off the top of your head, would you say you have a positive or negative image of each of the following?
Capitalism Socialism
The point of a poll like this is just to see how sick people are getting of capitalism.
I reject your distinction between social democracy and socialism. Whether the means of production are owned by the state or by a private citizen whose profits are almost all taxed away by the state isn't important enough to be declaring people ignorant.
If you want to distinguish between socialism and communism where all money and private property are abolished that is fine. People like Bernie Sanders are not confused about what socialism is so much as just not as socialist as they could be - for instance Bernie is against UBI.
Even the article you link doesn't agree with your categorization of socialism:
On the one side stood social democrats, who believed in using the power of the democratic state to reform capitalism. And on the other side stood democratic socialists, who believed that leftists should not do anything about capitalism’s crises because ultimately it was only through the system’s collapse that a better world would emerge.
I think this article does get more to the point though - can capitalism be reformed or do you need its collapse first? I think it can be reformed.