It's unclear where you got the impression that collective ownership of the means of production is a definition of socialism invented by the American Left. The idea originates with Marx and Engels, the latter elaborating that it is both necessary and sufficient for the elimination of social classes:
"...said Marx, what we observe historically is that only one class of persons has always owned or monopolized the means of production throughout history. This condition of ownership over the means of production is the single most fundamental fact of the materialistic theory of history since it is this that leads to the division of society into economic classes." [Morrison, Marx, Durkheim, Weber: Formations of Modern Social Thought, p. 44, paraphrasing Marx, The Germany Ideology, pp. 8-13.]
"...it was seen that all past history, with the exception of its primitive stages, was the history of class struggles; that these social classes warring with each other are always the products of the relations of production and exchange -- in a word, of the economic relations of their epoch; that therefore the economic structure of society always forms the real basis, from which, in the last analysis, the whole superstructure of legal and political institutions as well as of the religious, philosophical, and other ideas of a given historical period is to be explained." [Engels, Socialism: Utopian and Scientific, p. 72.]
"The proletariat seizes state power and to begin with transforms the means of production into state property. But it thus puts an end to itself as proletariat, it thus puts an end to all class differences and class antagonisms and thus also to the state as state." [Engels, Socialism: Utopian and Scientific, p. 93.]
tl;dr--In the analysis of Marx and Engels, ownership of the means of production is what divides economic classes; economic relations are the basis upon which are founded all social and political relations in society; in the struggle between those who own the means of production and those who do not, it is by the proletariat seizing the means of production that class differences are eliminated in society.