not exactly... you are closer to communism with that definition.
Agreed. But how is socialism not like that?
Socialism does not preclude private enterprise, you are just not allowed to operate sweat shops or otherwise unfairly exploit your workers or tenants if you are a landlord.
Socialism is central economic planning, central planning can only be done by government with force. Capitalism is a distributed (individual) economic system. "Private enterprise" in a socialist system means that the socialist system has allowed this particular private enterprise, which means it's still socialism. Central planning is in control.
That is what confuses liberals, you think as long as you allow any individual choice, you're not socialists. It's like saying that because you can turn left or right at an intersection, the streets aren't run by the government.