First of all, your definition is faulty. You are supposed to move the king and the rook with the same hand, and you are supposed to move the king first.
Aside from that ā
(āAside from that, Mrs. Lincoln, how did you enjoy the play?ā)
With a ānormalā (non-castling) move, the āpositionā while the pieces are in the air is of no importance. In the following position, if white plays Re1xRa1 ā
ā it does not matter that the white king is in check from the black rook while the white rook is in the air between e1 and a1. All that matters is the starting position, the ending position, and the nature of the move (was it ārook-likeā? Yes.).
With castling, itās a bit different, because of the special rule that you canāt castle through check. You almost need to view Ke1-g1 as a temporary move from e1 to f1 followed by another from f1 to g1. During the temporary stop at f1, the white king must not be in check.
Applying this logic to the Powerful Monarch position at the top of this post, we see that the white king is not in check during its temporary stop at f1, because at that moment it is defended by the white rook at h1.
I could have written my description of the castling maneuver to specify only one hand, and it would have been legal and would have posed the same problem.
(The illegality of using both hands, and the illegality of touching the rook first, both seem like odd digressions in the discussion of a fairy chess variant. Itās kind of like talking about the time control or the role of the TD.)
Reading over your reply, it occurs to me that the rule of original-chess to the effect that āyou canāt castle out of checkā does not apply to PM chess. If your rook can see your king, you arenāt in check, therefore you donāt have to worry about castling out of check. Moreover, of course you donāt have to worry about castling into check. So I suppose that castling through check could be excused, or maybe, just to be difficult, it could be made illegal. These rules werenāt handed down on stone tablets. Do whichever you prefer.
You donāt have to worry about castling through check, either, because your rook can still see your king as it moves through f1.
Please recall that having your king defended means it cannot be in check from your opponentās king, but it can still be in check from your opponentās other piece(s).