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[Full abstract in the paper] We aim to better constrain the atmospheric properties of the directly imaged exoplanet 51~Eri~b by using a retrieval approach on higher signal-to-noise data than previously reported. In this context, we also compare the results of using the atmospheric retrieval code \texttt{petitRADTRANS} vs a self-consistent model to fit atmospheric parameters. We present a higher signal-to-noise $YH$ spectrum of the planet and revised $K1K2$ photometry (M$_{K1} = 15.11 \pm 0.04$ mag, M$_{K2} = 17.11 \pm 0.38$ mag). The best-fit parameters obtained using an atmospheric retrieval differ from previous results using self-consistent models. In general, we find that our solutions tend towards cloud-free atmospheres (e.g. log $τ_{\rm clouds} = -5.20 \pm 1.44$). For our ``nominal'' model with new data, we find a lower metallicity ([Fe/H] $= 0.26\pm$0.30 dex) and C/O ratio ($0.38\pm0.09$), and a slightly higher effective temperature (T$_{\rm{eff}} = 807\pm$45 K) than previous studies. The surface gravity (log $g = 4.05\pm0.37$) is in agreement with the reported values in the literature within uncertainties. We estimate the mass of the planet to be between 2 and 4 M$_{\rm{Jup}}$. When comparing with self-consistent models, we encounter a known correlation between the presence of clouds and the shape of the $P-T$ profiles. Our findings support the idea that results from atmospheric retrievals should not be discussed in isolation, but rather along with self-consistent temperature structures obtained using the retrieval's best-fit parameters.