The concurrent societal goals and environmental challenges force industries to become more human-centric, sustainable, and resilient, which is envisioned as a value-oriented Industry 5.0 paradigm. The technology-driven Industry 4.0 paradigm is in a prosperous stage. Advanced information and manufacturing technologies have shown promising benefits, but challenges persist in the integration and implementation of smart manufacturing systems. Manufacturing system design (MSD) not only affects production efficiency and product quality but also is related to the strategic planning and sustainable development of manufacturers. New requirements for MSD appear and MSD has revealed new trends. MSD in the interplay of Industry 4.0 and Industry 5.0 is discussed in this two-part review, based on literature exploration within the Web of Science database. To obtain a systematic understanding of methodological, procedural, and technological exploration in a balanced way, this two-part review builds a Thinking-Modeling-Process-Enabler (TMPE) framework for reviewing MSD. In this paper (Part I of the two-part review), MSD methods are summarized and categorized from the design thinking viewpoint. Then, the manufacturing system modeling methods are categorized and discussed. Challenges and future research directions are identified in the MSD evolution towards Industry 5.0. Part II will detail the design processes and enablers (i.e., the P and E dimensions of the TMPE framework). This two-part review is anticipated to offer novel insights for advancing MSD research and engineering in the interplay of Industry 4.0 and Industry 5.0.