Depends on what definition you use, classically a social Democrat is someone who believes in using democratic means to gradually and peacefully transition from capitalism to socialism. The definition you are using is actually closer to a social liberal but even that isn’t quite right, welfare does not imply a mixed market economy on its own, welfare can exist in a pure laissez fair market economy. Social liberalism is in favor a mixed market economy which is a further step left from simply a regulated market economy with welfare. In a mixed market economy key industries are publicized and owned and operated by the government for the public benefit while allowing for the possibility of private competition as well as an open market economy in other industries and areas.
Uhm… Gonna have to wildly disagree with you there, wartime and postwar industrialization and strong unions created the American middle class, globalism breaking the unions is what has destroyed the middle class.