Proprietary trading can be better described by first looking into the role of a trader in an investment bank.
Using a large reputable investment bank as an example, it would have 4 to 5 traders covering U.S. interest rates products (Treasuries, agency debentures, etc).
These traders are there to make markets for their clients, but they also have their own positions and are allowed to use their balance sheet for such purposes. In that regard, they are like asset managers with market-making responsibilities.
Now, proprietary trading is purely using one’s balance sheet to make money for the firm (the traders are rewarded based on the pnl). The prop traders don’t offer any client services, and they exist solely as a risk taking group in the bank.
Just to reiterate, both client facing traders and prop traders have their own risk positions, but one is doing it “on the side” on top of what they do to make markets, but it is the full time job for the other.