You state that “PMI uses the term ‘process groups’ to refer to stages in the project life cycle, which appear as the columns in the below figure. Each column is therefore referred to as a ‘process group’. Each stage of the project life cycle is referred to as a process group rather than a stage or step in the life cycle.”

This is incorrect. The process groups are NOT stages, phases, or any part of a project life cycle. They provide a grouping of the project management processes described in the PMBOK Guide to facilitate their proper application. In a multi-phase project (i.e., one that is organized by a defined and understood project life cycle), the processes of the process groups are conducted iteratively within each phase of the project.