There's also nothing natural or inevitable about it, and it doesn't necessarily arise from owning property. There was plenty of private property around long before it was invented. As a matter of simple historical fact it took many years of effort and deliberate refinement to come in to bring.
I know supporters like to believe that it's as natural as the sun rising. But if you look at sheet amount of social displacement, violence and legislation that was required to make it a stable system, you'll see that's definitely not true.
As someone who's been using the property-based definition, I do think this one is more accurate. However, that would imply cavemen agreeing to scratch each others back would be practicing Capitalism (which I think they are).
More generally, that suggests that once two agents have the capacity for peaceful exchange of productive capacity, then Capitalism can occur. Capitalism is impossible in the world of reptiles, since they can't communicate or even internally model an exchange. Any service they provide to others, like defending their eggs, has to be hard-coded (i.e., forced upon them by nature).
The pure capitalists will argue that the alternatives to it are varying degrees of the use of force, to compel agents to either provide or not provide productive capacity beyond their internal desire. Whether that's necessary for a stable system is a different topic, but may be a function of the capacity we have to act as rational agents and how much we're still dominated by hard-coded instructions.
Speak to someone from the Soviet Union if you’re unsure of this.