No it's a totally legitimate political separation.
No it isn't. Newspapers, political pundits and newsmedia in general are known liars in the Authoritarian control structure, be careful which aspects of their culture you assimilate.
My fix to your breakdown:
Liberalism= Free association, private property, capitalism, minimal or zero government
Authoritarianism = Everything valuable owned by the elite, communism OR fascism, the proles are equally poor either way, maximum government
"Left" and "Right" are words that have no natural political meaning. Those words (Left & Right) are concerned with relative horizontal position from the perspective of an observer. All political meanings are projected, and using left-right as the polarities instead of up-down or red-blue is likely because of just how pliable the concept of left-right is (up-down and red-blue cannot change, whereas if you walk down a street one direction, what's on the left or the right changes if you walk it the other direction i.e. totally context dependent).
Like I said earlier in the thread, Left-Right is just a way of dividing people into 2 sides that can't win. The 2 sides should be uniting to fight against the actual problem: authoritarians who think they can live your life better than you can.