Galaxy major mergers are a potential mechanism for triggering active galactic nuclei (AGN) activity, but their role remains debated, particularly beyond the local Universe. We aim to shed light on the merger-AGN connection at z=0.5-2, exploiting the multi-wavelength datasets and James Webb Space Telescope (JWST) observations in the COSMOS field. We construct a stellar mass-limited sample and identify AGN via mid-infrared (MIR) colours, X-ray detections, and spectral energy distribution (SED) fitting. We train convolutional neural networks to identify mergers with mock JWST observations. We create non-AGN and non-merger control samples matching the redshift, stellar mass, and star-formation rate distributions of the AGN and mergers. We find AGN to be moderately more frequent in mergers than in non-mergers, with excess ratios ranging from ~2.5 (X-ray AGN) to ~1.3 (MIR) and ~1.1-1.2 (SED AGN). Similarly, AGN galaxies show a higher merger fraction (f_merg_) than non-AGN controls. We then study f_merg_ as a function of relative and absolute AGN power, utilising the AGN fraction (f_AGN_) and accretion disc luminosity (L_disc_) parameters. We uncover a f_merg_-f_AGN_ relation with two regimes: f_merg_ stays roughly flat for less-dominant AGN (f_AGN_<0.8) but increases at f_AGN_>0.8 for the MIR and X-ray AGN, and more gently for SED AGN, where mergers appear to be the main triggering mechanism. Additionally, f_merg_ increases monotonically as a function of L_disc_, for all AGN types, reaching f_merg_>50% for the most luminous AGN (L_disc_>~10^46^erg/s). Overall, our results suggest that major mergers can trigger AGN out to the cosmic noon at z~2. Furthermore, the role of major mergers shows a clear dependence on AGN luminosity and remains the principal mechanism for fuelling the most powerful AGN.