A market economy is the method by which resources are allocated, by the individual decisions of buyers and sellers.
Capitalism is better as a description of ownership of resources. The definition you quote is good, but can be misleading ( one could confuse resource allocation with resource ownership ).
In other words, one could see a centrally planned capitalist economic model. Unfortunately, examples to date include Japan ( and by extension, Germany and Italy of the 1930's )
Additionally, people tend to mistake the US as a pure market-capitalist system. It is not. The US has a mixed economic system.
My sin was to take a definition of "capitalism" out of a dictionary. That's wrong because "capitalism" is not a term that needs a definition, it's a concept that needs an understanding. My Econ 1A prof used to describe communism as "state capitalism" because the capitalist market mechanisms of supply and demand were being used to allocate resources, it was just that the state (and not the individual) was setting the prices.
Just like the protectionists that want the state to control prices with import taxes in order to decide who gets what.