On one hand, we have Marx' definition, characteristics of an economy and polity: urbanism, bourgeois elites, private property, wage labour. It was good foresight. The world he described di more or less come about. More than at any other time we work in companies, for wages, in cities. Most of the important property is privately owned.
On the other hand, we have anglo-american ideological "capitalism." It's a variant of liberalism. The emphasis is on free markets, and a confidence that these produce an efficient self organizing market. But, the ideals (at least of the philosophers they refer to) are a little bit of awkward fit in 20th & 21st century settings, where the economy is made up of massive companies.
Adam Smith, the American founders, Locke, Hume... and others that form the basis of ideological capitalism... they idealized independent people, usually farmers. Entrepreneurs, maybe 10 person tin factories. Facebook or Unilever... Idk if an economy of mostly these is all that capitalist, by the proponents' definition.
Anyway, I think the ideological lens (neoliberalism, asian-capitalism) might be more obscuring than clarifying. In practice, the economic systems are the same. The ideological differences between the US & China do not impact the stucture of the economies more than (for example) geography or local cultural norms do. The Soviet centralised economies were different, but I'm not really sure how related that is to ideological communism. It's just what they did when all those commune projects failed.
..The actual differences are more in the legal systems and the political norms (which in China are still extremely strict). Here I think we agree.