Viewing it entirely in economics is incorrect. All of the above can be done under capitalism. The key difference is not what form of economics are employed but which class controls power and puts the resources of the state to use.

The capitalist state is a state where capital owners hold power and use that power to exploit more capital.

The socialist state is a transitionary state in which the workers have seized power and use the state to repress the bourgeoisie and put resources to their own use.

The communist state is what occurs when capitalism is entirely defeated, all nations are socialist, conflict is eliminated and material abundance is achieved, at which point states start to stop existing as the resources within them that are put towards repressing the bourgeoisie through violence are put towards other things when there is only 1 class in society.

Okay, so extremely abridged, here is what seperates capitalism from socialism.

Under capitalism, private individuals own the means of production, distribution, and sustenance. Workers are forced to go to one of these private individuals and exchange their labor power for a wage. Capitalist profit is generated by paying the worker less than their labor power is worth but enough to sustain workers as a class. The workers are prevented from using the means of production without entering into the wage labor model through the threat of physical violence.

Under socialism, the means of production are managed in common, somewhere along a sliding scale of the people working in a workplace and democracy having control of how the workplace operates depending on the system

You'll note that these both can operate within markets, and both require at least some planning.

Video: we need a mixture of capitalism and communism is bullshit

Book: Explaining why markets are bad

Edit: this is ignoring the way the state plays a role in these economic formations but Im trying to keep it simple.

For the record, I think before this your definition of capitalism was defensible, but then communicating clearly would require using the term "liberalism" to describe the government.