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 Cu0.7Zn0.3 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.