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Next-generation weak lensing (WL) surveys, such as by the Vera Rubin Observatory, the Roman Space Telescope, and the Euclid space mission, will supply vast amounts of data probing small, highly non-linear scales. Extracting information from these scales requires higher-order statistics and the controlling of related systematics such as baryonic effects. To account for baryonic effects in cosmological analyses at reduced computational cost, semi-analytic baryonic correction models (BCMs) have been proposed. Here, we study the accuracy of a particular BCM (the A20-BCM) for WL peak counts, a well-studied, simple, and effective higher-order statistic. We compare WL peak counts generated from the full hydrodynamical simulation IllustrisTNG and a baryon-corrected version of the corresponding dark matter-only simulation IllustrisTNG-Dark. We apply galaxy shape noise matching depths reached by DES, KiDS, HSC, LSST, Roman, and Euclid. We find that peak counts from the A20-BCM are (i) accurate at per cent level for peaks with S/N < 4, (ii) statistically indistinguishable from IllustrisTNG in most current and ongoing surveys, but (iii) insufficient for deep future surveys covering the largest solid angles, such as LSST and Euclid. We find that the BCM matches individual peaks accurately, but underpredicts the amplitude of the highest peaks. We conclude that the A20-BCM is a viable substitute for full hydrodynamical simulations in cosmological parameter estimation from beyond-Gaussian statistics for ongoing and future surveys with modest solid angles. For the largest surveys, the A20-BCM must be refined to provide a more accurate match, especially to the highest peaks.