You are highlighting meaningless semantic differences and narratives in place of meaningful structural properties to define the two.
"The means of production" is not a relevant idea outside of marxism. It's just called capital, assets or money to everyone else. Defining capitalism in terms of the "means of production" is part of a dishonest attempt to reframe the discussion in Marxist terms which is not taken seriously by anyone that is not Richard Wolff in academic economics and not by anyone who is involved in economic analysis professionally.
Check out the wiki for the "means of production". It's part of the series on marxism (ie economics used by essentially no one in practice): https://en.wikipedia.org/wiki/Means_of_production
Focusing on the word private is not untrue, it's just dishonest in the sense that it is used to imply that any other meaningful framework exists and again to try to highlight the comparison to Marxism as primary which is not a debate that exists on any meaningful implementation level. Out of over 200 countries in existence, as far as I know North Korea is the only one potentially that doesn't acknowledge private ownership of goods (on paper only) and that only exists for unrelated complex international political reasons, not because it is a tenable system economically.
You don't own things privately if there is no market to trade them in because markets are an emergent phenomenon of people owning things privately. The only way there wouldn't be this market is if you legally restricted private ownership. So market's and private ownership are meaningless without each other.
And even then you are failing to understand the difference between markets as a concept and capital markets.
So no, it is not a mistake to equate markets with capitalism, especially the existence of capital markets and this does not preclude markets existing before we thought of things in terms of capitalism.
Also Matt Stoller is not an economist, he's a partisan author of popular media. He's not authority on any of this.