a few corrections.

Marxism is the belief that the productive capacity of human civilization should be directed at meeting the needs of humanity directly.

Marxism isn't the belief that humanity "should" do anything. If it does, then it would fall into the realm of moralistic ideology. As I said before, Marxism is morally neutral. Marxism is an analysis of society and a movement to "abolish the present state of things."

Marxism makes a distinction between what is called personal property and private property. Personal property are things that one person could reasonably be expected to own and utilize.

This is way too vague of an abstraction. Marx believed personal property was private property, but that personal property is property you've earned through your work. For example, if I make a chair, then that chair is my personal property because I made it. Or, in a capitalist society, if a worker earns a money (a wage) for their work and exchanges that wage for a tv. He's earned that tv through exchange.

Private property would be a mine or a factory or a farm, the things that produce all the goods that modern society is based upon.

no those are means of production. also, they don't "produce" the goods. workers do by laboring upon the means of production.

Marxists believe that private property should be owned collectively to produce goods for all.

Marxists are proponents for the abolishing of private property.

Here's a quote from my favorite Marxist philosopher, Syndrome:

When nobody owns the means of production, everybody does.