Market impact corresponds to price moves due to the intensity of trading in one direction. As you mention: when a market participant trades in size in one direction, the price usually follows.

They are two reasons:

1. the first one is informational: the trader is right to buy (or sell), he or she understood the price will go up (or down). Hence in this case, a relation with volume is not needed: the price will move.
2. the second one is mechanical: it is the one you mention. The pressure of the buying flow moves the price. In such a case the relation between the traded quantity $Q$ and the price move $\Delta P$ is square root (Bacry, Emmanuel, Adrian Iuga, Matthieu Lasnier, and C-A L. "Market impacts and the life cycle of investors orders" Market Microstructure and Liquidity 1, no. 02 (2015)): $$\Delta P\propto \sigma\sqrt{Q\over V} + a\,\psi,$$ where $\sigma$ is the volatility of the specific instrument, $\psi$ its bid-ask spread, and $V$ its traded volume.

JP Bouchaud explain the two effects quite well in his Introduction to Part II. Price Impact: Information Revelation or Self- Fulfilling Prophecies? to Capponi, Agostino, and C-A L, eds. Machine Learning and Data Sciences for Financial Markets: A Guide to Contemporary Practices Cambridge University Press, 2023.

Market impact corresponds to price moves due to the intensity of trading in one direction. As you mention: when a market participant trades in size in one direction, the price usually follows.

They are two reasons:

1. the first one is informational: the trader is right to buy (or sell), he or she understood the price will go up (or down). Hence in this case, a relation with volume is not needed: the price will move.
2. the second one is mechanical: it is the one you mention. The pressure of the buying flow moves the price. In such a case the relation between the traded quantity $Q$ and the price move $\Delta P$ is square root (Bacry, Emmanuel, Adrian Iuga, Matthieu Lasnier, and C-A L. "Market impacts and the life cycle of investors orders" Market Microstructure and Liquidity 1, no. 02 (2015)): $$\Delta P\propto \sigma\sqrt{Q\over V} + a\,\psi,$$ where $\sigma$ is the volatility of the specific instrument, $\psi$ its bid-ask spread, and $V$ its traded volume.

JP Bouchaud explains the two effects quite well in his Introduction to Part II. Price Impact: Information Revelation or Self- Fulfilling Prophecies? to Capponi, Agostino, and C-A L, eds. Machine Learning and Data Sciences for Financial Markets: A Guide to Contemporary Practices Cambridge University Press, 2023.