This is how the labels are taught in modern Political Science:
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These categories are just ways of grouping methods of governing. They are only as useful as they are distinct. Socialism is often used by the right to denote anything that involves the government redistributing wealth. This however, is not a useful definition of socialism in practical reality. Because among the governments that redistribute wealth, there are massive differences.
In particular, conservatives wish to group all forms of socialism with the early to mid 19th century kind practiced by Stalin. This socialism was very authoritarian and did not remotely live up to any high minded ideals American socialists had. It is also very different from modern democratic socialism countries, which would probably put directly in the middle of this cross. But nevertheless, it allows them to successfully demonize modern democratic socialism -- which is really what the goal is.