This is a good summary. To simplify/describe it slightly differently, the definitions of capitalism and communism and socialism decribe the relationships between workers and the value of the work that they do.
Under capitalism, owners own the business and employ workers who are paid wages. The value of the products or services that the workers make is more than they are paid - and this extra value produced (profit) goes back to the owner. Thus, you end up with two classes of people, the workers/wage laborers (proletariat in communist jargon), and the owners (bourgeoisie in communist jargon).
In socialism, the workers own the businesses, and so the profit goes back to them instead of to the non-working owners. Thus in socialism, there is only the one class, the workers, and the bourgeoisie class has been removed.
This is extremely simplified, but I think describes the overall work and value relationships under the two systems.
I hope this helps too.
Aren’t all anti-communist activities money laundering schemes?