They're right though. Freely trading goods and services is commerce, and an economy based on commerce is a market economy.

Capitalism is specifically a market economy where the means of production are owned privately by an investor class, who increase their wealth exclusively by "profit": a discrepancy between the value the workers in a company contribute, and the compensation paid to those workers (Price- Cost [including wages] = Profit).

There are non-capitalist market economy models. Like market socialism, in which economic production is still generated by private companies in a free market, but those companies are owned by the workers themselves rather than by non-employee shareholders.