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Some feedback on the most recent edit(s):

• From the looks of things, it seems like the pieces that capture by pushing/pulling are (in game-mechanics terms) performing rifle captures that only work when the target piece is on (or at least near) the land/sea border, in such a position that the push/pull word make it move over that border.
• Some of the pieces have familiar names, but don't move like their familiar versions. I'd recommend either putting a note to that effect at the top of the listing, or adjusting the pieces to follow tradition (both are equally valid, and I figure you'll go with the former).
  • The usual Centaur is a Man (non-royal King) + Knight (aka Horse) (KN). What you describe here (fRbB) is usually a Hunter -- which, now that I think of it, would be a valid piece for this game (in connection with Demeter).
  • The more familiar form of the Medusa moves like a Queen, or leaps two spaces (QS). Still, there's not really any reason that, for this game, the Immobilization wouldn't also be possible.
  • I think I went into Nereids, Pegasi, and Hippogriffs already.
• That said, as much as I'd like to see the Sphinx combine the moves of lion and eagle (even if it's just as simple KNS[F-R]), it's still a recent invention of my own, so I'll understand if you end up with something else.
• I could be mistaken, but a "drops anywhere after capture" feature on the Phoenix may be a bad thing. I personally would limit it to the Phoenices' starting spaces, or at least the player's own territory.
• Some clarity is needed on the Harpies' move. If they can only swap one square away, then it can't be "across territories." On the other hand, I suspect that you meant that they capture enemy pieces one square away; presumably they can do their swap at any distance, as long as it's across two different territories. And, presumably, the two territories have to be of the same type unless the switched piece is a Sky Army piece. (BTW, "swapping" has two P's.)
• About the Muses: if a piece is captured, it probably doesn't matter if its color is changed. Perhaps a more appropriate way to describe what they do is to give them a Grasshopper move (landing on the empty space beyond the target piece) and/or Advancer move (landing on the empty space just short of the target) which causes the target to change color.
• The crushing capture seems like it's either a costodial capture (like a Pincher, capturing a target by having a friendly piece on the opposite side) or something very similar (if the capturing piece has to displace the target).
• To clarify my earlier descriptions, in case you didn't get this: A Carrier is a rarely-used type of piece that literally "carries" another piece. The carried piece uses a "capturing" move on the Carrier, and the Carrier uses its own move to go to where the carried piece needs to go. Once there, the carried piece then uses its own move to leave the Carrier. From a conceptual standpoint, any type of ship or mount could be a Carrier.
• Having to place both Kings in checkmate is an interesting twist, and one I don't think I've seen before (though it wouldn't surprise me if certain individuals could find a half-dozen CVs that already have it). A few things to consider:
  • If one King is checkmated and the other is being pursued, can the defending player do something to free the checkmated King (like capture the threatening piece)?
  • Can the checkmated King be captured before the other one is checkmated? Might that even become the actually requirement (capture one and checkmate the other)?

A more general thought: What if one were to conceptualize the board as being an array of nine separate boards? Each of the nine territories is already the size of an orthodox chess board, so emphasizing the difference visually could be as simple as putting an extra board underneath the continents (thus raising them above the water). It wouldn't change the actual play in the slightest, but it might help the player to visualize what's going on.