You are very close. Government owned does mean its owned by the people who control the government. 100%. But who controls the government? In the west, government is largely controlled by the wealthy (industry lobbying is an easy example). So then how do we have a government that is controlled by the masses?

This is where the type of government called a socialist government comes in. Communism, on the other hand, is some distant future where humanity no longer has to worry about scarcity, and all ideas of money and state are gone.

So a communist party is named thusly to say "we wish to work towards that goal, but recognize that there are steps required to get there". For example: increasing the democratic control of a country, both politically and economically, while decreasing the influence of capitalists, whose interests are in contradiction to the interests of those who work for a living (or need to sell their labour to make a living, since labour is the only commodity most of us control).