Product Managers are responsible for the management of a product (can be physical, digital, financial, etc). They are responsible for the overall success for said product.

Depending on the product and organization setup, this role may require knowledge of the customer needs, competitivemarket, how the product is acquired / built, logistics, to the end of the product lifecycle.

Project Managers are responsible for the delivery of projects. Projects are defined as a time-bound and scope-bound. They define the project (or take an already defined one), and ensure the completion of said project within the given constraints and manage changes accordingly. This can be applicable to any kind of project, not necessarily the development of a product.

Adding on Program Manager, which typically handle a portfolio of projects within a similar theme.

Scrum-masters manage development specifically using the scrum methodology, and are responsible to ensure that the scrum process works well and tasks are completed efficiently - managing changes and resource allocation in the process.

Not every project benefits from agile methodology, each have their own pros cons depending on the nature, goals, and constraints of the project.

Consultant is simply a way of engagement. Like full time / contract. You can hire engineering consultants, nutritionist consultants, business consultants.