I do not think there is a specific “person” who leads design as referring to who in the question. There are so many different design companies that I think each individual one leads their own. In electronic companies, they control the design of levels and every aspect of the game. These companies can have separate design producers considered the lead designers. The people that control the design of levels of each game lead their own design specifically for the layout of the levels such as candy crush. For web designers, there can be up to three or more designers that lead in producing just one website. It usually takes a team of designers to complete a properly functioning and professional website as the one my dad hired a company to design for his business. My cousin who is an interior designer does not lead designs because she works for a company. She can assist with design ideas for the hotels they are assigned. In that case, the owner of her company leads the designs because the final plans for the hotels depend on her opinion. If you are working in the architectural design business, I think each architect should lead/design building projects. The owner also leads design because they control what the architect’s finished project and design looks like.