"Every economic system in the modern world has elements that are both capitalist (private ownership and / or administration) and socialist (public ownership and / or administration)." - Open Left
This polarizing, misleading, and ambiguous definition of socialism rests on several major misunderstandings:
1) The difference between socialism and capitalism is 100% objective.
1a.) There is no "intention difference" between the two systems.
2) Less government guarantees more capitalism.
3) The wealthy are the automatic enemies of the public good.
I see no basis for any of these assertions. Let's examine a few other systems:
Feudalism - In Feudalism, cousin of fascism, a noblemen is entitled to rights over a vast area of land and all the people living on it in exchange for millitary service. Money may or may not be involved at all, but in any case, you won't have a right on how you earn it or spend it, as you, like 95% of the population, will have one job you are given that you must keep for the rest of your life or face execution. Like socialism, government control is universal, even to the point of government slavery, but like capitalism, private property exists and most decisions at the top tier are profit-maximizing.
Anarchosocialism - In Anarchosocialism, no government exists, but no private property exists either. The mob, the posse, serves as the only source of law and order, but redistribution is universal and total. This is based off the philosophy of Karl Marx, but by Open Left's definition of capitalism, is the ultimate capitalist society, as here, even the police are privatized. Can't get more private then the gunhand! (If you are interested, a good Anarchosocialist blog can be found here.)
Fascism - In Fascism, government regulation of businesses reaches the point where they can only serve the state interest. All of society, from the top to the bottom, is geared to serve those in power, and what private property does exist is largely nominal. Some attempt at redistribution is made, but inequality in a large sense is both tolerated and adored. Some businesses are government-owned, but still traded as stocks. Some would consider this system very socialist for it's mammoth level of government control, others very capitalist for it's emphasis on corporate power, efficiency and elitism, and some towards the center. Either way, resort to the public/private distinction and your mind will be twisted in to a slinky when gazing at it, as here the two blur in to nothing.
Tribalism - A tribal society largely shares resources, but still tolerates a strong man who can be seen as the "CEO" of the group. Trade is very free and almost corporate in it's conduct often times, but nearly everything is at whim. Where does it fit?
I think these are enough examples to illustrate my point. Now, back to homework.