  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.
