That is not how economics describes socialism. Economics is more complex than your explanation.
That elements of government are not socialism in your definition?
What you are talking about is the ABSOLUTE or EXTREME definition which simply never exists.
There is no such thing as ABSOLUTE, 'Capitalist', 'Democratic', 'Free Market', "Socialist' or ANY OTHER system people may live under.
They all would fail the true "economic definition' test as they all have some elements of other systems.
Republiclowns know that as they decry ELEMENTS only within america as Socialist while there are clearly other elements that are Capitalist and others that fit other labels.
So once we except REALITY and that all societies will have a mix of these elements then it becomes a game of checking boxes, and deciding which elements we want in each box.
That is how you define democracy. This collective decision on which what will fit in to each box and be delivered by which ELEMENT.
What Republi'cans' are now doing is saying 'everything i disagree with is wrong and bad and socialist' and everything i agree with 'is good and right and not socialist'.
So you can have two different Republi'cans' who might disagree on certain elements from 'Policing' to 'Firefighters', to 'Teachers' to etc, etc, etc, on which ones are OK for the gov't to forcibly take your money and provide service for, and they each call the other 'Socialist' because of the area they disagree on. But as long as they agree they say 'that is not socialist because we agree'.
It is a stupid way to think and to view things but then they are republi'cans'. So stupid is just their thing.