Although Cu/ZnO-based catalysts have been long used for the hydrogenation of CO₂ to methanol, open questions still remain regarding the role and the dynamic nature of the active sites formed at the metal-oxide interface. Here, we apply high-pressure operando spectroscopy methods to well-defined Cu and Cu_0.7Zn_0.3 nanoparticles supported on ZnO/Al₂O₃, γ-Al₂O₃ and SiO₂ to correlate their structure, composition and catalytic performance. We obtain similar activity and methanol selectivity for Cu/ZnO/Al₂O₃ and CuZn/SiO₂, but the methanol yield decreases with time on stream for the latter sample. Operando X-ray absorption spectroscopy data reveal the formation of reduced Zn species coexisting with ZnO on CuZn/SiO₂. Near-ambient pressure X-ray photoelectron spectroscopy shows Zn surface segregation and the formation of a ZnO-rich shell on CuZn/SiO₂. In this work we demonstrate the beneficial effect of Zn, even in diluted form, and highlight the influence of the oxide support and the Cu-Zn interface in the reactivity.