First of all: Since we're talking birds, have you ever heard of the Flying Kingfisher?? It's my favorite fantasy chess piece, or at least the one with the best name.

for the 4 listed birds...

• Firebird (A) is nominally Man plus Knight plus Camel, minus the ability to jump. Man plus Knight is already very close to Dragon, which is valued around 15ish (a case could be made to drop it to 14, but that's neither here nor there). Similarly, base Fortress is currently 16 and this is probably about similar in power. Overall I would expect Firebird to cost 15ish.
• Icebird (B) is also Man plus Knight but adds Zebra (3,2) paths instead of Camel paths. Although they control the same amount of territory, more of Icebird's squares are forward-pointing, which I view as slightly more positive than the sideways defensive nature of Firebird. Overall Icebird could be either 15 or 16; I suspect some would cost it at 16 and I think I would agree.
• Lightning Bird (C) is Knight plus Ferz plus Camel-path. This piece seems obviously weaker than Firebird due to the missing Wazir spaces, but some value gets added back for the ability to jump. The holes in Lightning Bird's control area make it feel more similar to Archbishop than Fortress, even though it's a more Rook-shaped movement style. Whether this piece or Waterbird below is more valuable probably depends on whether a Knight appreciates the addition of Ferz or Wazir more. For now I'd stick it at 12 and see if it needs a nerf later.
• Waterbird (D) is the easiest to cost: it's a Knight++ with some extra (3,2) pathing. This addition feels similar to the extra value provided by Legionary+ over base Legionary, although we're adding 8 squares instead of 4 and they're move/attack squares rather than just attack only. Maybe a more apt comparison would be to current Valkyrie+++, which is 14, although Waterbird can't snipe the back row from the 5th the way Valkyrie can, and the King teleports must also be worth something. Overall, Waterbird feels like a 12-13 point piece, leaning more towards 12.