What is the radical difference between ‘this’ and ‘a’ when telling a story?

I think that the word emphatic was poorly chosen. The intention is to give your story a sense of immediacy; suddenly it's not about people or objects in the abstract, but (nearly) concrete individuals, e.g. not just

"Cedric did a weird thing (that I'm going to describe to you)"

but

"Cedric did this weird thing (that you are witnessing as well at the same time as me because I am relating the story as though it is happening right now)."

(thanks to @Kit.)

Of course, it doesn't really work; the story doesn't really become any more real to your listener, and if you over-indulge in this technique the constant repetition of "this" starts to be a bit annoying. For this reason, most people (of my US English-speaking acquaintance and reading, at least) stop talking this way after high school; writing dialogue that uses this technique is a good way to establish your characters' age. Definitely "informal."

Use 'this' when you're referring to something specific:

• 'I use this kind of potion' [to point toward a particular variety of potion]

Use 'a' when you are speaking more ambiguously:

• 'I use a kind of potion' [to speak of potions more broadly]

In other words, if you're speaking of a set, use 'a'; if you're speaking about a particular element of that set, use 'this'.

This seems like an attempt at informal speech. It isn't meant to emphasize anything, except perhaps the way normal people speak.

And the characters are normal people (within the context of a Harry Potter story).

This only happens in informal speech, so there are no written down rules of grammar for this usage. I can tell you what I think, though, but I don't have any references to back me up and I may be completely wrong, so feel free to downvote me. My impression is first that this is typically used at the start of stories. Here,

Cedric did this weird thing ... And that Fleur girl tried this sort of charm ...

are both the starts of the sections of the story about Cedric and about Fleur. The this here is used to indicate that a new topic is being introduced.

Second, it's used for emphasis; it emphasizes the thing it modifies. So here it indicates that "this great jet of flame" was something impressive.

You can, of course, overuse the technique by replacing nearly every a with this.