Nope. That's not at all what those terms mean. Capitalism is not "when competition," and socialism is not "when no competition."
Capitalism = private ownership of means of production
Socialism = collective ownership of means of production
Capitalist monopolies are privately owned. A monopoly is no less privately owned than a minor company in a sector with healthy competition, and there is nothing remotely collective about a monopoly. A lack of competition does not in any way mean that "the people" own the organization. Not theoretically, and not actually.
Collectively owned socialist enterprises are not necessarily free of competition. In fact, heated competition over quotas had a lot to do with many of socialism's worst blunders.
Competition does not remotely guarantee that costs are low by international standards either.
