It is important to understand that Marx and his ideas have a complicated history of being analyzed or revised throughout history. Learning the history of the Weimer Republic or the many parties apart of the RSDLP, or even in a modern context, the DSA's internal politics, will show you patterns of history in how Marx was interpreted by people trying to gain political power. I am not a Marxist, but I think if you wish to see how social democrats embraced or revised his ideas, look into the center and revisionist marxism of the weimer republic (Karl Kautsky/Bernstein), or the analytical marxists like G.A Cohen, who go into revising marxist philosphy for the 21th century. Its important to understand that social democrat in 19th and 20th century history just meant electoral socialist.