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Next-generation weak lensing (WL) surveys, such as by the Vera Rubin Observatory's LSST, the $\textit{Roman}$ Space Telescope, and the $\textit{Euclid}$ space mission, will supply vast amounts of data probing small, highly nonlinear 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 BCMs 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 expected at the depths reached by DES, KiDS, HSC, LSST, $\textit{Roman}$, and $\textit{Euclid}$. We find that peak counts in BCMs are (i) accurate at the percent level for peaks with $\mathrm{S/N}<4$, (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 $\textit{Euclid}$. We find that BCMs match individual peaks accurately, but underpredict the amplitude of the highest peaks. We conclude that existing BCMs are 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, BCMs need to be refined to provide a more accurate match, especially to the highest peaks.