I like this, I've found designing Ways to be great fun. Find great mechanics and/or describe animal behaviour at the same time. I'll put my list of ideas here and comment on Segura's as I go.
My version of Way of the Eagle is Spy:
Way of the Eagle
+1 Action
Each player (including you) reveals the top card of their deck and discards or puts it back, your choice. Then draw a card.
For when you want to know your top card or your opponents. And unlike bears, eagles can fly over moats; hopefully the Attack is mild enough. Maybe this shouldn't draw a card either, to limit the temptation to play Spy chains that got the 1st edition card removed. But then is it strong enough?
I thought of a Way of the Wolf that has the same premise as Fox, counting copies in play, reflecting its being a pack animal and working nicely with Wolf Den. Not sure if + $1 per copy is a bit too strong? A 5/5 split on a Village or Lab is quite an easy + $5. I had:
Way of the Wolf
Gain a card costing exactly $1 per copy of this you have in play.
Could be too hard to pull off nicely most of the time, but maybe fine on a Way.
For Fox I had scavenging:
Way of the Fox
+1 Buy
You may put your deck into your discard pile.
A niche effect, but when you do want it it's useful, so being on a Way lets you access it at those times easily. +Buy helps usefulness more, and vaguely reflects the fox on Ranger.
Hippo is nice and can be strategic. Magpie is obvious if not weak in comparison to the Pig; 'Otherwise you may discard it' maybe, to move Victories along. Manatee would have the sideways card that played as it and the set aside card left out for next turn, wouldn't it? Quite a sacrifice.
Surely Octopus could be a Vulture. I have this on my ideas list:
Way of the Vulture
Gain a non-Victory card from the trash. If it costs $2 or less, + $2.
Limited to non-Victory to avoid Province tfb, and has the boost for $2-s for when tfb isn't available.
Speaking of Octopuses though, here's an idea I had that could fit:
Way of the Spider
If you have exactly 8 cards in play, gain a card costing up to $5.
Octopus might be the friendlier animal to choose for the general public. Gaining a $5 might be too nice a reward, it could tone down to + $3 or something.
Panda is also doing the counting cards in play premise, feels like it should be nice.
(Deleted Way of the Parrot)
Reindeer is nice, a limited Peddler. Saves a collision of two identical terminals.
Tiger is an outtake Donald mentioned in his Menagerie secret history. Option overload, analysis paralysis.
For the others, why not? Well I'm not too sure on gargoyle taking away all the limitations exorcist imposes (buying it, lining the two cards up).
And here's a few more ideas:
Way of the Shark
Trash up to 2 cards from your discard pile.
Could be a hermit crab, after Hermit, but sharks eat stuff underwater i.e away from your hand. This follows the same idea as my fox, narrow windows of opportunity that are easily accessible as a Way. 2 cards might be too many, too swingy.
Way of the Rabbit
If this is the second time you've used Way of the Rabbit this turn, gain a copy of each card that followed the Way.
Another premise: count the number of times the Way is used. This could go bigger than 2 for something exciting. Don't know about balance here.
Way of the Tortoise
+ $2
Put a card from your hand onto your deck.
Go slow, retract into shell. Sometimes useful terminal clash solution.
Thrones could indeed be too strong for a Way, so how's this?
Way of the Crow
Once per turn: gain a Curse to play this twice.
Once per turn saves this endlessly throning itself. And did I see this posted somewhere else?
These are just ideas, no testing yet. I've only had a little go with the Shark, nothing conclusive. Again though, nice thread!