Socialized medicine is not really a technical term so much as it is a pejorative term for health care plans that conservatives do not like. However, the general meaning of socialized medicine is health care where the government pays for people's health insurance. Because the government pays for the health insurance, it also has a great deal of say over what kinds of medical procedures are covered and how much providers are paid for the procedures. In that way, it is socialized because it is government-controlled rather than controlled by the free market. (By this definition, though, Medicare is socialized medicine yet people rarely refer to it that way because it is a popular program.)
This is often contrasted with a direct fee system in which people would pay for the medical care that they receive. They would decide for themselves what care they would buy based on their needs and the price of the service. The government would not be involved in these decisions.
In looking at the links, please note that "americanprogress.org" is a liberal site and keep that in mind as you read what is said there.