Testing is an information or intelligence-gathering activity performed on behalf of (people who I will call) testing stakeholders. The manager who asked you to test could be your most important stakeholder (ask them!) They think testing is important enough to get someone as important as you involved – but might not be able to articulate why they see it as an important role.
You might use the axioms to help you have a rational dialogue with them, to home in on the important issues you need to resolve to enable you to do the right job.
By the way, if you are testing the products of your own efforts, you could be your own stakeholder. Your approach to testing your own products or systems will be focused on what you and others, as stakeholders, want to learn about those products or systems.
Most systems that need testing have stakeholders whose interests do not coincide perfectly. We cannot test everything, so we need to help them to make choices. We need to develop a good relationship with stakeholders to build consensus, buy-in and trust in our test approach. Since most stakeholders are non-technical, the language we use must be simple and direct. The Test Axiom definitions are just that.