There is no being coy at all, I can't even twist my head around how these similar places in the US and Russia are not the same. What is his "alternate definition" where such a place in Russia is stripped of the word market?
That's a minor point as you answered the question, sort of. But not to satisfaction. A market in the US or USSR sells commodities at a set price. People buy goods or they don't. The markets operate exactly the same.
As you're attempting to make a rational argument unlike others, I will get to the point. The markets in the US and USSR were the same, despite you following some US ideological line and calling the US one a "free" one. It is production that was different, and control of production. Misdirecting the difference from control of production to "free" markets is just more misdirection and sophistry (which is widespread, not particularly pertaining to you).
It is perfectly reasonable to use the term "market" that way. After all, what else would you call a place where radishes are stored with price tags? But that's not what the person arguing with you is saying, and trying to rebut them based on an alternate definition is unproductive.