correct definition of communism in Marxism

I am far from being an expert on Marxism. But my impression is that what you say is at best an oversimplification. For instance, in the Communist Manifesto I find no claim that nothing should be called communism unless it successfully offers limitless plenty to all. I do find things like this:

In this sense, the theory of the Communists may be summed up in the single sentence: Abolition of private property.

and this:

The Communists are distinguished from the other working-class parties by this only: 1. In the national struggles of the proletarians of the different countries, they point out and bring to the front the common interests of the entire proletariat, independently of all nationality. 2. In the various stages of development which the struggle of the working class against the bourgeoisie has to pass through, they always and everywhere represent the interests of the movement as a whole.

You will notice that nowhere in that second quotation does he add "3. Communists believe that they will successfully bring about an end to all forms of economic scarcity.". (So: Marx may perhaps have believed that an end to scarcity was inevitable, or something of the kind, but he doesn't appear to have thought that such a belief is a requirement for someone to be called a communist.)

I don't (of course) deny that an end to scarcity was a goal of communism. For that matter, it is (or should be) a goal of capitalism too. And Marx, fond as he was of the idea that his preferred system was a matter of historical inevitability, may well have believed -- or at least found it useful to say -- that when communism is fully implemented scarcity will be at an end. But none of that is the same as saying that 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.