First, let me point out that there is a terminological mess here. I was careful to specify that I was talking in Marxist terms which do not match terms used in contemporary Western political discourse. Note, by the way, that we are not talking about theories (one of which is named "communism"), but about forms of society.

Marxists call "communism" a particular form of future society which has never (yet) been realized. It's an aspirational form, the carrot in front of the donkey, the light at the end of the tunnel, the heaven in which the worthy will find themselves. The realized, intermediate form is called "socialism". The USSR was a socialist country.

In Western political talk, "communism" and "communist" refers to real societies like Soviet Russia and Communist (!) China, while "socialism" means a capitalist state with a generous welfare system, e.g. Sweden.

In any case, a Google search will give you lots of Marxist definitions of communism. Let me quote you Wikipedia to start:

A communist economic system would be characterized by advanced productive technology that enables material abundance, which in turn would enable the free distribution of most or all economic output and the holding of the means of producing this output in common. In this respect communism is differentiated from socialism, which, out of economic necessity, restricts access to articles of consumption and services based on one's contribution.

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if you abolish private property, social class, etc., and this regrettably fails to bring about a total end to scarcity, then what you did wasn't communism after all.

It wasn't. The Russians, for example, who abolished private property, etc. did not call their society "communist". They called it "socialist" and said that they are only building communism.