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Satellites in core level spectra of photoelectron spectroscopy (PES) can provide crucial information on the electronic structure and chemical bonding in materials, particular in transition metal oxides. This paper explores satellites of the Ti 1$s$ and 2$p$ core level spectra of SrTiO$_3$ and TiO$_2$. Conventionally, soft x-ray PES (SXPS) probes the Ti 2$p$ core level; however, it is not ideal to fully capture satellite features due to its inherent spin-orbit-splitting (SOS). Here, hard x-ray PES(HAXPES) provides access to the Ti 1$s$ spectrum instead, which allows us to study intrinsic charge responses upon core-hole creation without the complication from SOS and with favorable intrinsic linewidths. The experimental spectra are theoretically analyzed by two impurity models, including an Anderson impurity model (AIM) built on local density approximation (LDA) and dynamical mean-field theory (DMFT), and a conventional TiO$_6$ cluster model. The theoretical results emphasize the importance of explicit inclusion of higher-order Ti-O charge-transfer processes beyond the nearest-neighboring Ti-O bond to simulate the core level spectra of SrTiO$_3$ and TiO$_2$. The AIM approach with continuous bath orbitals provided by LDA+DMFT represents the experimental spectra well. Crucially, with the aid of the LDA+DMFT method, this paper provides a robust prescription of how to use the computationally cheap cluster model in fitting analyses of core level spectra.