The reason why this “colloquial definition” is this way is so that capitalists can convince the masses that capitalism is natural “because it has always existed” by claiming that antique slave society, feudalism and even late hunter gatherer society were actually capitalist. This isn’t a neutral definition that is as valid as the other, it is a lie crafted for propaganda purposes and shouldn’t be taken seriously.
What no theory does to you.
No seriously, you need to read on this, you clearly have at best a very simplistic understanding of the subject.
Private property and markets can’t just be abolished immediately after a revolution, it’s not magic. Young socialist systems have to go through a transitional phase during which private property and markets are still allowed under strict oversight of the state.
His does not make them capitalist as the proletariat still has control over this private sector via the socialist state, such as in China where all of the essential industry that is necessary for every other, known as the commanding heights, are fully state owned and the enterprises that are private are required by law to have a party member on their board as well as a “golden share” owned by the state that allow it unchallenged veto power over the board’s decisions among other means of authority over the private sector.