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It can be classified anyone of them depends on the person position. For me, it is a consulting profession.
Project Management in most cases are like contractors, and their lent of employment depends on the type of product they are dealing with. Sometime they are part of the Executive team, which in that case they review most of the projects that are proposed internally or externally.
All of them and more from initiation to hand over including planning, execution and mentoring within budget, and according to schedule and quality.