It would be good to clarify the role of product owners outside the Scrum team which seems to be just as critical. The product owner would need to define product strategy including discussion with customers. I would also think they are involved in marketing and sales as well. Ultimately as the owner, they are responsible for the success of the product.
Somehow, the description in the guide seems to focus just on the development side (sprints and features) which does not give the full picture of the role. In my view a Product Owner is in fact a Product Manager as well.