Exactly markets aren't the distinction, communist and socialist democracies all have markets. A really interesting model of that was Allende's Project Cybersyn in Chile before the US sponsored fascist coup that put Pinochet in charge. There's highly regulated markets within capitalist countries as well, bulk energy is largely very "designed" and regulated markets.

The Marxian view of socialism would consider it as a transition state between capitalism and communism. While someone may be ideologically communist, they will likely have more political opportunities catering to socialist policies in capitalist democracies with a "left" party. Revolutionaries don't believe this is possible, and argue capitalism's structure won't be threatened by socialist policies unless a revolution occurs, and might even consider comrades who support socialist parties as "not real" communists. Germany's socialist party supporting ww1 is often used in forms of this argument.

Ultimately in a lot of these capitalist democracies, there are individual leftists but no real political power, this is certainly the case in the US. Working to raise class-consciousness and labor organizing is basically the front of whatever left exists there. It's a bleak time to be on the left, and sometimes I wish I could have the enthusiasm of the self-righteous liberals who naively think that if everyone regardless of identity was distributed equally in the capitalist system everything would be right and fair.