The guide is says pretty much what i say. "How this is done may vary widely across organizations, Scrum Teams, and individuals."
OTOH, The humane, real-world understanding of the word "product" and the word "owner" coming together should prime over whatever definition there is, because none reads definitions except for the purpose of an argument ;-)
No, i'm not even sure if you're arguing here in good faith. The next sentence after the one you cited clearly states with examples:
"The Product Owner is also accountable for effective Product Backlog management"
