I see Betza's notation as a convenient and intuitive way of describing the moves of a chess piece. I believe XBetza's notation has another goal, that of being able to be understood by an AI for the cases not covered by the basic notation.
In principle there is no limit to what an AI can understand; it is just a matter of how much effort you put in creating it. And I subscribe to the goal of making move description convenient and intuitive, even for moves that are intrinsically more complex, such as the Griffon, Falcon or Chu-Shogi Lion. So think it is worth it to put a great deal of effort in achieving that goal.
But for any system of notation it seems important that it is unambiguous, and covers sufficiently many cases. Betza himself has suggested the t[F,R] notation for Griffon, but this is not unambiguous, as an 'Acute Griffon', which would continue inward instead of outward, making a 135-degree turn, (or the piece that can continue in all four Rook directions), would be written the same. Furthermore, the leading t seems redundant here: the 'then' meaning could be implied by the presence of the brackets. I think Betza suggested also other uses for the brackets, like z[F,W] for a slider that alternats W and F steps in a crooked way, but this distinction could also have been made by using other separators than comma for that, e.g. [W/F]. Once non-alphanumeric characters are allowed in the notation, lots of options are available.
I hope you agree that it hardly matters for the intuitive understanding whether one would write F&R, F-R, [F-R] or [F,R]. But now suppose that we want to slam extra W moves on a Griffon... I don't like WF&R so much, because intuitively you would parse this as (WF)&R, as punctuation usually separates 'words'. While it is intended to mean W(F&R). This is why I think it would be better to require some sort of parentheses around the complex moves. And to stay as closely as possible to Betza's legacy, these can be brackets. With W[F-R] there is little doubt that the W is a separate move.
The ambiguity problem has to be solved by putting some directional specification on the continuation legs, to distingush outward from inward, or even between several possible outward directions (such as in the case of the Ship). In your example you suggested a new prefix character ($, as we have run out of letters) for indicating 'outward'. This has the advantage that we can keep using the original directional modifiers in the meaning they have in the absolute frame of reference. This works well for the Ship, where you can write F&$vR to make the extra v pick out the vertical of the two outward Rook paths.
But I wonder whether introducing a whole new system for indicating relative directions is worth it. For one, the use of an absolute direction to disambiguate equivalent relative directions only rarely works; the Ship is a favorable case. It would for instance not work on a 'Chiral Griffon', which would bend always in the same relative direction (left or right), to give a rotation-symmetric rather than reflection-symmetric move pattern. And how would you know that in N&$R 'outward' would mean 'in the direction of the long component'? Some are v and others s so you cannot use those to indicate it. And if you define the meaning of 'outward' on oblique moves as in their long direction, how would you describe a piece that would continue like a Rook in the short direction of the knight move? It seems that we would have to create a whole new system for describing relative directions, or groups of directions, similar to the existing Betza system for absolute (initial) directions, but using puntuation characters instead of the familiar fbvlrs. That would involve a lot of new characters, but perhaps these could be chosen in a somewhat intuitive way, so they would not be too hard to remember (like ! for outward, ^ for inward, < for turn left, > for turn right, _ for < or > and | for ! or ^). But it still seems a hassle, which can be exploited only very rarely.
So I think there is a lot of merit in sticking to the existing Betza scheme for indicating directions, but interpret them as relative directions on continuation legs. Then instead of needing a new symbol like $ or ! for 'outward', we can simply use f for outward. In a sense the Betza directional modifiers have always been relative: for simple moves they are relative compared to the way the player is looking. We also say fmWfcF for black Pawns! (And in a 4-player game even v and s have player-dependent meaning.) Is F&fR really any less intuitive than F&$R ? You would lose the possibility to disambiguate the two outward Rook moves by use of an absolute direction, but you now have a simple way to write a (right-handed) Chiral Griffon: F&frR. One always has to compromise: what makes one piece easier to describe, will make some other piece that behaves in the opposit way more difficult to describe. I think it would be a mistake to judge a description system solely on how well it does on a single piece that you happen to use, but is not very common, and even less representative for the general problem.
There also is a more fundamental issue: writing the Griffon as F&fR or [F-fR] assumes the move can also be terminated without making all its legs, after just the F step. It might be subjective whether this is intuitive or not; it seems to me that this is mainly inspired by the case at hand (i.e. bent slider). Problem is there are many other cases where you would not want this. E.g. lame leapers like the Mao also need a multi-leg description to indicate where they can be blocked. But if W&fF or [W-fF] also allows making a W move, how can we describe the Mao then? And if fcF&fmF would also allow plain capturing diagonally forward (fcF), how would we describe the Checker? We do need a way to describe complex paths that cannot be terminated half-way. One way would be to forbid incomplete moves, so that these have to be described as separate moves when they are desired. (Advantage: no new notation needed. Disadvantage: move descriptions get longer.) The alternative is to introduce a different symbol for continuations that are only optional. E.g. [F~fR] could mean a Griffon that can make F moves, as the fR continuation is optional. While [F-fR] could not make the F move, although it can be blocked on the F square, as the fR continuation is mandatory. Actually I sort of like this latter solution, even though it requires introduction of a new symbol.
I would also like to have some feedback from other people, on what they would consider intuitively clear in these matters.
In principle there is no limit to what an AI can understand; it is just a matter of how much effort you put in creating it. And I subscribe to the goal of making move description convenient and intuitive, even for moves that are intrinsically more complex, such as the Griffon, Falcon or Chu-Shogi Lion. So think it is worth it to put a great deal of effort in achieving that goal.
But for any system of notation it seems important that it is unambiguous, and covers sufficiently many cases. Betza himself has suggested the t[F,R] notation for Griffon, but this is not unambiguous, as an 'Acute Griffon', which would continue inward instead of outward, making a 135-degree turn, (or the piece that can continue in all four Rook directions), would be written the same. Furthermore, the leading t seems redundant here: the 'then' meaning could be implied by the presence of the brackets. I think Betza suggested also other uses for the brackets, like z[F,W] for a slider that alternats W and F steps in a crooked way, but this distinction could also have been made by using other separators than comma for that, e.g. [W/F]. Once non-alphanumeric characters are allowed in the notation, lots of options are available.
I hope you agree that it hardly matters for the intuitive understanding whether one would write F&R, F-R, [F-R] or [F,R]. But now suppose that we want to slam extra W moves on a Griffon... I don't like WF&R so much, because intuitively you would parse this as (WF)&R, as punctuation usually separates 'words'. While it is intended to mean W(F&R). This is why I think it would be better to require some sort of parentheses around the complex moves. And to stay as closely as possible to Betza's legacy, these can be brackets. With W[F-R] there is little doubt that the W is a separate move.
The ambiguity problem has to be solved by putting some directional specification on the continuation legs, to distingush outward from inward, or even between several possible outward directions (such as in the case of the Ship). In your example you suggested a new prefix character ($, as we have run out of letters) for indicating 'outward'. This has the advantage that we can keep using the original directional modifiers in the meaning they have in the absolute frame of reference. This works well for the Ship, where you can write F&$vR to make the extra v pick out the vertical of the two outward Rook paths.
But I wonder whether introducing a whole new system for indicating relative directions is worth it. For one, the use of an absolute direction to disambiguate equivalent relative directions only rarely works; the Ship is a favorable case. It would for instance not work on a 'Chiral Griffon', which would bend always in the same relative direction (left or right), to give a rotation-symmetric rather than reflection-symmetric move pattern. And how would you know that in N&$R 'outward' would mean 'in the direction of the long component'? Some are v and others s so you cannot use those to indicate it. And if you define the meaning of 'outward' on oblique moves as in their long direction, how would you describe a piece that would continue like a Rook in the short direction of the knight move? It seems that we would have to create a whole new system for describing relative directions, or groups of directions, similar to the existing Betza system for absolute (initial) directions, but using puntuation characters instead of the familiar fbvlrs. That would involve a lot of new characters, but perhaps these could be chosen in a somewhat intuitive way, so they would not be too hard to remember (like ! for outward, ^ for inward, < for turn left, > for turn right, _ for < or > and | for ! or ^). But it still seems a hassle, which can be exploited only very rarely.
So I think there is a lot of merit in sticking to the existing Betza scheme for indicating directions, but interpret them as relative directions on continuation legs. Then instead of needing a new symbol like $ or ! for 'outward', we can simply use f for outward. In a sense the Betza directional modifiers have always been relative: for simple moves they are relative compared to the way the player is looking. We also say fmWfcF for black Pawns! (And in a 4-player game even v and s have player-dependent meaning.) Is F&fR really any less intuitive than F&$R ? You would lose the possibility to disambiguate the two outward Rook moves by use of an absolute direction, but you now have a simple way to write a (right-handed) Chiral Griffon: F&frR. One always has to compromise: what makes one piece easier to describe, will make some other piece that behaves in the opposit way more difficult to describe. I think it would be a mistake to judge a description system solely on how well it does on a single piece that you happen to use, but is not very common, and even less representative for the general problem.
There also is a more fundamental issue: writing the Griffon as F&fR or [F-fR] assumes the move can also be terminated without making all its legs, after just the F step. It might be subjective whether this is intuitive or not; it seems to me that this is mainly inspired by the case at hand (i.e. bent slider). Problem is there are many other cases where you would not want this. E.g. lame leapers like the Mao also need a multi-leg description to indicate where they can be blocked. But if W&fF or [W-fF] also allows making a W move, how can we describe the Mao then? And if fcF&fmF would also allow plain capturing diagonally forward (fcF), how would we describe the Checker? We do need a way to describe complex paths that cannot be terminated half-way. One way would be to forbid incomplete moves, so that these have to be described as separate moves when they are desired. (Advantage: no new notation needed. Disadvantage: move descriptions get longer.) The alternative is to introduce a different symbol for continuations that are only optional. E.g. [F~fR] could mean a Griffon that can make F moves, as the fR continuation is optional. While [F-fR] could not make the F move, although it can be blocked on the F square, as the fR continuation is mandatory. Actually I sort of like this latter solution, even though it requires introduction of a new symbol.
I would also like to have some feedback from other people, on what they would consider intuitively clear in these matters.