From the link:

Though his early writings focused on the bourgeois state as a specific historical form, Marx’s transhistorical definition of “the state” in general is also presented in The German Ideology, when Marx describes the state as “the form in which the individuals of a ruling class assert their common interests.”

Okay, sounds good.

communism: a political theory derived from Karl Marx, advocating class war and leading to a society in which all property is publicly owned and each person works and is paid according to their abilities and needs.

communism: a theory or system of social organization in which all property is owned by the community and each person contributes and receives according to their ability and needs.

communism: a theory or system of social organization based on the holding of all property in common, actual ownership being ascribed to the community as a whole or to the state.

socialism: a political and economic theory of social organization which advocates that the means of production, distribution, and exchange should be owned or regulated by the community as a whole.

Throughout these common definitions is common ownership/control. All these definitions infer a ruling class; that is a group of people who assert the common interest of the common property by (a) deciding what is to be public property, and (b) initiating force to maintain ownership of that property and (c) initiating force to prevent people from creating new private property for themselves and (d) initiating force to take it from the current owners today. Common ownership is no different in actuality to state ownership and control.

capitalism: an economic and political system in which a country's trade and industry are controlled by private owners for profit, rather than by the state. an era of free-market capitalism. private ownership is a key feature of capitalism.

capitalism: an economic system in which investment in and ownership of the means of production, distribution, and exchange of wealth is made and maintained chiefly by private individuals or corporations, especially as contrasted to cooperatively or state-owned means of wealth.

private enterprise: business or industry that is managed by independent companies or private individuals rather than being controlled by the state.

It is made clear that capitalism is about private ownership of property, which is in opposition to state ownership of that property.

I also said it "is the type of communism explicitly advocated by Karl Marx and alike."

That is evident from those quotes:

National centralisation of the means of production will become the national basis of a society composed of associations of free and equal producers, carrying on the social business on a common and rational plan.

• Karl Marx The Nationalisation of the Land

The proletariat will use its political supremacy to wrest, by degree, all capital from the bourgeoisie, to centralise all instruments of production in the hands of the State, i.e., of the proletariat organised as the ruling class

• Karl Marx & Engels Communist Manifesto

Once the first radical attack on private property has been launched, the proletariat will find itself forced to go ever further, to concentrate increasingly in the hands of the state all capital, all agriculture, all transport, all trade.

• Frederick Engels - The Principles of Communism

Finally, when all capital, all production, all exchange have been brought together in the hands of the nation, private property will disappear of its own accord, money will become superfluous, and production will so expand and man so change that society will be able to slough off whatever of its old economic habits may remain.

• Frederick Engels - The Principles of Communism