That's pretty oversimplified. If you want use a simple way of categorizing socialists, then I would say you need to have
at least the
libertarian-authoritarian scale, but you should also have the revolutionary-gradualist scale (see the article on the early history of
social democracy).
On the far Left, you have two groups, namely
Libertarian Socialism (aka left-libertarianism, Anarcho-Syndicalism being the most "pure" form of that ideology) and Authoritarian Socialism (aka state socialism, with Communism being the most "pure" form of that ideology). Historical examples of Anarch-Syndicalism are
Revolutionary Catalonia and the
Free Territory of Ukraine, and the historical examples of Communism are, of course, the USSR and China, amongst others.
The revolutionary-gradualism is where the term "social democrat" vs "revolutionary" arises. This is the disagreement over how to obtain a socialist society: Through gradual reforms over time or through a large revolution at once. However, more recent terminology simply refers to an acceptance of capitalism and concentrated control of private capital, but with some checks in place and social safety nets. (Therefore, this is this is "Social Democratic" state capitalism vs "Neoliberal" mixed capitalism vs Libertarian private capitalism --but these are all different arguments amongst a capitalist configuration of society, so there's no attempt at removing an amalgamation of private power and control by unaccountable individuals in the system).
No. Social Democracy was originally about whether the progress of socialism should be slow or fast. In the mid-twentieth century, social democracy became a "let's only reform capitalism, and not accept socialism" ideology, as the Wikipedia article indicates.
The notion that a "liberal" is a type of "socialist" is not merely a literal contradiction of historical terms, it's not even close to being true in modern political parlance. Some socialists will call themselves liberal (as an American political term for meaning "I'm on the Left"), but, for instance, I do not.
Scandanavian countries are capitalists who accept the Social Democratic "state/mixed capitalism" model. In other words, they have a strong central state that controls portions of the economy, like healthcare/social safety nets/education/academic research, but they are capitalists. They don't object to there being individuals who control huge segments of the private industry.
Some people will call this socialist anyways (e.g. Bernie Sanders, Jeremy Corbyn, Labor parties across the West, etc), and I don't fully object to this. But we should be careful to note that it's always dangerous to have massive amount of private capital accrued in the hands of a small number of people. The failure to deal with this is what's lead to the borderline-if-not-outright corporate oligarchy and the spread of neoliberalism across the US, UK, and EU for decades now, and the weakening of social safety nets everywhere.
