If 'All' of you have agreed and established a shared understanding, I would ask, do we need a dedicated document? Most teams capture it as part of their 'Definition of Done' which is usually Big and Visible to everyone.
I've seen teams create test strategy document where someone wanted to have it and others simply had to follow it. Obviously this document resided on a SharePoint site and hardly anyone looked at it. On the other hand, I've worked with teams who worked in regulated environment, following SOX, there the test strategy didn't exist as a separate document.
If you are still pressed to create this as a separate document, I would ask myself - Do the developers have a Development Strategy Document ? what about the Product Owner - does she have a dedicated Product Ownership Strategy document where she's listed the way she'll come up with ideas, refine, validate and manage ideas? No. I recall how one Scrum Trainer (CST) explained a similar question that when we go somewhere, we don't just forget to put our clothes on. Why? Was it because it was documented somewhere? Because right from childhood we learned it and shared the same understanding with others in family and society. Now it has become common sense.
Cheers!