"Classical Liberalism" started off as Adam Smith's economic liberalism, religious toleration, and removal of legislated social hierarchy. In the 19th century progressive political thinkers in Britain, John Stuart Mill being the most well known, further developed liberalism based on the suffering and misery that economic liberalism brought to the working class. The basic premise was that the government should regulate the economy only so far as necessary to preserve the guarantees of the social contract. Still pro-capitalist, but with labor laws and public education to ensure the liberty of all.
Socialism (as a coherent political ideology) developed around the same time, but in places that lacked the representative political structures that Britain had which allowed it to respond to the problems of industrialization through legislation. Without that representation, workers in France and then Germany began preaching a more extreme version of Mill's progressivism, which sought to eliminate all inequality and all competition. Socialism itself then split into Social Democrats who thought this could be achieved through legislative action (as continental European governments become more representative after the 1848 revolutions). Communists, on the other hand, thought that once you begin working within the "system" then you become corrupted by it, and continue to perpetuate the class inequality that you were trying to solve. Marx, therefore, saw the only way to achieve "Scientific Socialism" was through violent revolution, and that social democrats were just as bad as Classical Liberals.
Mill's "progressive" liberals, then, were fighting for very similar ideas as the Social Democrats, just with slightly different philosophies. Those slight differences have become simplified over time to the point where your average American can't differentiate between a Social Democrat and Communist, let alone a Social Democrat and a Progressive Liberal.
