'Illegal position' is not something that is defined in the FIDE rules, or anywhere else I am aware of. There just are illegal moves, and those are moves that violate at least one game rule. You could of course define an illegal position as the position that arises from an illegal move, but they are not always recognizable: if I start a FIDE game with 1. Nxh8 that would definitely be an illegal move, but the same position could also have been reached through legal moves, so it cannot be an illegal position. Castling through check or out of check is an illegal move in FIDE, but the resulting position can be fine.

There is a universal restriction on all pieces to not make illegal moves. That is what 'illegal' means. What actually is legal and what not depends on the detailed game rules. In some games it is illegal to castle when your King is attacked, in others it it is illegal to capture a Lion with a Lion when the latter is protected, in some games it is illegal to capture an Emperor with an Emperor when the latter is protected. In some CVs it is illegal for Kings to stand next to each other, in other CVs there is nothing against it, or they are not even allowed to face each other from a distance. There is nothing universal about that.

So no, I don't think we are talking about different things at all. Pseudo-legal moves are moves that a piece has considering the occupation of the departure square, target square and possibly the squares along the path that it is supposed to take to get to the latter. Like in move diagrams. What is on the board outside the path taken in one of the possible moves has no effect on the pseudo-legality of the latter. (Even if it is in the path of other moves the piece might have!) For an orthodox King the pseudo-legal moves would be the adjacent squares not occupied by a friendly piece, for an Emperor it is the entire board except squares occupied by friendly pieces. (There is no path, as it is a leaper, so the pseudo-legality of each move only depends on the departure and destination square, just as with the King.) There just are some additional game rules that might cause a pseudo-legal move to be in fact illegal, such as that you cannot go there if the the square is under pseudo-legal attack, unless the enemy King is on it. Or that you cannot go there if the square is under pseudo-legal attack and there is an Emperor there.

The rules for game-terminating moves can seem different from what they are in the pre-termination part of the game, e.g. after a King strolls into check (illegal!) you are allowed to capture it with a pinned piece, while during the game that would never be possible. But that doesn't mean they have to be different in a universal way. Each CV can specify its own rules for the after-moves. In fact the difference with the pre-termination rules can often be made to disappear by just having rules for the number of after-moves granted in various situations ('normal' move, capture of royal, baring a royal, reaching a goal square). But then that number of after-moves can arbitrarily vary from CV to CV, there is nothing universal about that. And a different number of after-moves translates into seemingly different rules for the (after-)moves that preceded them.

We are spending way too much attention on Tai Shogi, which was just one example (and because of its multiple royals, a rather complex one). Better focus on Recursive Chess, where King capture is not legal when it exposes your own King to capture on the next move.