You only gave part of the definition. Capitalism requires private property and profit. That's because the source of capital is profit. Look up "capital accumulation."
Capitalism Define Capitalism at Dictionary.com
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.
Where do you see the word "profit"?
Capitalism - Definition and More from the Free Merriam-Webster Dictionary
: a way of organizing an economy so that the things that are used to make and transport products (such as land, oil, factories, ships, etc.) are owned by individual people and companies rather than by the government
Where do you see the word "profit"?
Profit is a natural result of capitalism, because people tend to not invest their money without making a profit as a goal. A farmer invests seed into the ground, with the profit of a large harvest. I invested money into education, with the profit of getting a hire paying job. I invested money into a car, for the profit of getting to work.
The fundamental definition of Capitalism, does not require a profit. Profit is what naturally happens when you can control your own investment.
Socialized systems have profit too. It's just a profit of the state, instead of a profit to the individual.
If a state run company makes no profit, the result is disaster. The difference is, in a capitalist system, the lack of profit for a company is a disaster for the company. In a socialized system, lack of profit is a disaster for the entire country, because the entire country pays for it. Think Fannie and Freddie as an example.