What you describe in the first sentence is plutocracy, rule by the wealthy elite, the exact opposite of communism. Authoritarianism is political, not economic and has nothing whatsoever to do with communism.
So communism in the PRC is an exception?
http://www.bloomberg.com/news/articles/2012-02-26/china-s-billionaire-lawmakers-make-u-s-peers-look-like-paupers
The net worth of the 70 richest delegates in China’s National People’s Congress, which opens its annual session on March 5, rose to 565.8 billion yuan ($89.8 billion) in 2011, a gain of $11.5 billion from 2010, according to figures from the Hurun Report, which tracks the country’s wealthy.
And isn't there enough evidence by now that the Bolsheviks were financed by the wealthy elite in the west?
What other real life communist rulers did you have in mind?
and according to dictionary.com ( I cant speak for its accuracy), communism describes both the economic and political
http://dictionary.reference.com/browse/communism
a system of social organization in which all economic and social activity is controlled by a totalitarian state dominated by a single and self-perpetuating political party