This kind of false reduces the diversity economic ideologies into basically just "left" vs "right". Even amongst leftist ideologies, there's communism, democratic socialism, syndicalism, market socialism, and probably a bunch more.

The most common way to categorize economic ideologies I've seen is along the lines of who owns which means of production: land (including natural resources), labor, and capital.

Socialists generally want social ownership of land and capital, and sometimes labor (depends on which socialist you ask).
Capitalists generally want private ownership of land, labor, and capital.
Georgists (like myself) want social ownership of land, but private ownership of labor and capital.

But even those don't reveal the complexity of how to implement social ownership of thing, or degrees of ownership. Income taxes, for instance, are a form of partial social ownership of labor, something both capitalists and many socialists on paper oppose.

Market socialists want a market economy but with worker-owned coops.
State socialists want direct state ownership.
Georgists want social ownership of land through hefty land value, externality, and severance taxes.

But even that ignores the why of the various ideologies and their proposed policies.

Syndicalists want to achieve greater equity through unions everywhere and in every industry because they believe the power of collective bargaining will level the negotiating power between employer and employees
Market socialists want a market economy full of worker-owned coops because they see markets as an incredibly powerful tool, and because they see worker-owned coops as a very robust and equitable form of enterprise within a market economy.
Georgists want to tax land and externalities and finite natural resources (as well as lower barriers to entry for free enterprise) because we believe rent-seeking and monopolism to be the source of rampant inequality, and that a truly fair and competitive market without market failure is the best engine for shared prosperity
Genuine capitalists think the inequality is simply the unavoidable price to pay for all the prosperity, and that inequality itself is a motivator to be productive

And of course not all people hold their economic ideologies for noble reasons. Plenty of (if not most) self-described capitalists hold those beliefs either out of greed (because they're greedy rent-seekers whose land/externalities/natural resource rents ought be taxed away imo) or complacency (they don't like upsetting the status quo). There are also tankies who believe in communism not because they truly want a fair and equitable society that respects human rights but because they're just a fascist reactionary who likes to feel morally superior to others. I'm sure there are also some market socialists or Georgists or syndicalists as well who hold their beliefs for non-noble reasons.

But anyhoo, all that to say there's a tremendous complexity to economic ideology, and I think it's best to not frame everything as simple left vs right, black vs white, us vs them.

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