Is mp something different than the moving and hopping modifiers combined? Does mpafsK mean the same as N? If so, how does it work?

I now elaborated on that, and hope this will make it clear.

I don't follow what the x modifer does. An example would be helpful.

I think we have to discuss presentation strategy here. The list of modifiers below the diagram also serves the purpose of a quick reference guide, which can always be kept in view by people who are using the sandbox diagram. Elaborating on each of them, explaining examples, would easily quadruple its size, so that it becomes difficult to use for that purpose. I do in fact give an example for x, (mNxaN) but I don't say what it is. (It is in fact the Knight of Knight-Relay Chess, which moves (without capturing) like a Knight (mN) and induces Knight moves at a distance of a Knight move (xaN)).

Would it be an idea to add an extra section where each of the examples mentioned in the 'quick reference' are discussed in depth?

It's not clear how FyafsF is supposed to describe the Griffon. Should the second F be an R?

Did you read the 'more about a' link in the 'quick reference'? It opens a box that should make this clear. I presented it this way, because most people wanting to use XBetza (e.g. for making an Interactive Diagram for their variant) would not need the complexity of multi-leg moves, and be completely satisfied by combining Knights with Bishops, or perhaps make a HFD. Even the need for directional modifiers is quite rare (outside Shogi). Not many people use bent riders, bent lame leapes, locusts, bifurcators, pieces that induce moves in other pieces, or displace those...

I don't want to burden the 95% of people with stuff they don't need, and deter them by it. The a is a highly advanced feature of XBetza, almost never needed.

But it would of course be bad if I hid it so well that people will miss it alltogether. The more-about-a box was actually a remnant of the old sandbox diagram I posted as a comment to the Betza article (since then pushed far in the background noise). In a real article I have more space, and perhaps I should move most of what is now in the box to the preceding section about XBetza above the sandbox, or a new section about multi-leg moves. (With an initial warning that people who just want to do basic things can skip that section.)

To answer your question: The yafsF part of FyafsF means the move starts as F to an empty square, to fork there (y is "move and continue as slider"), and then continue in the (relative) fs direction, i.e. as a Rook. That is a Griffon, except for the F moves. (Because a Rook makes at least one step; it cannot null-move.) So the latter are slammed on separately.

I'm not familiar with Bifurcators, and other readers might not be too. An illustration to how the piece works or a link to a page on this type of piece would be helpful before describing what gabyabsR  is doing.

Bifurcators are an invention of Mats Winther. But when I looked for them, I could not find any mention of them in the Piececlopedia. (Do we have a search function here?) I know Mats describes many variants on his own website, mostly based on very innovative piece designs ('Catapults' are another invention of him).

Note, however, that the sandbox diagram is available as an interpretation aid: to know what it means you only have to copy-paste it in the Betza move-entry, assign it to a piece, (e.g. 'betza2'), and then click on the name of that piece to summon up the move diagram. As this is some kind of hopper, even the move diagram will be interactive for pointing out the hopper's mount by hovering the mouse pointer over a square in its path (or touching that square on a touch-screen device). That would make it obvious what the piece does; as they say, a picture is worth a thousand words.

One reason for giving the examples was to provide people with something to try out the sandbox diagram with something non-obvious.

[Edit] I now added a section at the end, which discusses the examples of the more complex pieces in detail.