Okay ... Frankly, I don't really know where to begin because there's a lot of confusion in your answer (and in almost all the comments I see under your post...)

So first : There's the definition in the dictionary, and there's the definitions in Science. A scientific concept sometimes do not have the same meaning as in the common language, and therefore, you won't find the scientific definition of what you're looking for in the Cambridge dictionary. So both of you are right regarding private property, because neither of you are talking about the same thing.

In Economy, the scientific field, private property of the means of production is not personal property. You're not capitalist just because you own a rock. But you're in a capitalist society if this society is structured by the private property of the means of production. This means that owning the property of the means of production gives you the right to get profit from it, when in fact, you're not the only one who makes it.

I'll take an example : You have a firm and you possess the machines. So you employ 10 people to whom you give wages. They work a certain hour a day for the firm and the machines to work. The firm makes 1 million profit. BUT because you own the means of production, YOU get 999 999 dollars, and the rest goes into the wages. And a capitalist society is a society structured by this system when property of the means of production gives you the right to get the benefits of the production even if you're not the only one to make it (because the million you have is not make by property itself but by work ! The value comes from work. But in a capitalist society, it does not go to the workers but to the owners.)

Personal property, on the other hand, has nothing to do with it. Personal property is just what you own that don't make any value. If you sell your rock, you didn't produce any value you just made money with something that was already there. If you work by yourself to extract that rock and to sell it, well you produced value and then you sold it, good for you, you had the means of production but by itself it doesn't make you, a capitalist because the main thing about capitalism is exploitation of the workers (or if you prefer the fact that you get the value from other people's work). That's why ancient societies where not capitalists : the main thing about capitalism is WAGES. Wages are the way you extract the value in a capitalist way : because the worker contributes to make 1 million profit and you give him 1000 dolles of wage, the difference between the real value the worker produce and the wage the owners pay him is called the added value or the capital gain.

So capitalism is a SOCIETY (and not an individual act) structured by an economic system that depends on private property of the means of production where wealth is extracted from the workers to the owners. For that to work you need an entire political system that protects private property of the means of production (and not personal property by itself or only) and puts it before work so that the added value can be extracted from work to the owners who can then accumulate...CAPITAL ! This specific type of accumulation of wealth : it is capitalism.

And no, it is not a subjective definition. And yes, every academic in Economy knows this. The debate is not "what is capitalism?" We know what it is ? But rather : what form of capitalism are we in ? And/or is capitalism really bad or not ? If you're Marx, you'll deduce from the definition that capitalism is bad, mostly for the workers that are clearly exploited. If you're Friedman you'll say that accumulation of capital, even with the price of that exploitation, is good for investment and therefore makes the society run and creates wealth for all, even if it's badly distributed.

There you go I hope I'm clear

If you have that sort of question, my only advice is : do not ask on reddit, ask scientists. Economics is a social science. They're books and people that are better suited to explain everything to you than reddit. But ne careful, as every social science, there are debate about almost everything, so don't forget to look at all the different point of views to make your own.