I do agree, the center would be Dumbledore convincing a grieving Snape to protect Harry. It is also a powerful and memorable scene that stands by its own. Snape unimpressed with first-year Harry is not that memorable or not even important, and I think that's an important feature of the scene that ocuppies the place of the center of the ring.
About the title, it could be a hat tip to Chaucer, but it isn't the first time JK Rowling uses that format for chapters: #20 in the Order of the Phoenix is "Hagrid's Tale" too.