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An oxidizing agent undergoes reduction by accepting the formal Electron inventory. In the zinc-copper displacement experiment, ${\mathrm{Cu}}^{2+}$ plays this role.^[ref]

A reduction decreases an oxidation state, recording gain in the formal Electron assignment. For ${\mathrm{Cu}}^{2+}+2e^{-}\to \mathrm{Cu}$, the decrease is $+2\to 0$. The reduced chemical species serves as the oxidizing agent in the coupled redox.^[ref]

In the electrochemistry examples here, reduction is gain of Electrons by a chemical species; it occurs at the cathode. That electrode definition holds in both a galvanic cell and an electrolytic cell; it does not fix terminal polarity.