It seriously turns me off when organisations are unable to differentiate between a role and a job. Talking about this confusion, Scrum Product Owner probably one of the most misunderstood roles these days which unfortunately many organisations advertise as a job.
A job describes what one does daily. A role encompasses how our actions help an organisation to succeed. Next time you are not sure, ask yourself how you add value to the organisation.
Let me dumb it down for you. As a father, I play several roles with my daughter. When I am helping her to explain concepts of mathematics I am playing the role of teacher. When I am playing football with her, I am being a coach. In a nutshell, my job is parenting and as a parent, I assume the roles of teacher, coach, and many others.
So repeat after me - product owner is role one play on the scrum team and product manager is a job. As a product manager, you play several roles business analyst, UX designer, product owner, strategist, marketer, and so on.