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I think you’re confusing social democracy with democratic socialism. The first is as you say, and has huge overlap with liberalism, left liberalism, and progressive liberalism.
The second is achieving socialism through democratic means, without the need to overthrow government as once was believed to be entirely nessecary.
But then again, terms do change context over time, and by place. So maybe I’m the one who is wrong.
I guess we’ll see.