In the tips and notes it says "Note: It's up to the individual speaker to decide whether animals are animate enough to earn a B-class demonstrative pronoun." so you can basically choose whether they are animate or not. Hope that helps :)
It seems from the usage in the lessons that it's not animate vs inanimate and more person vs object/animal (like 'it' vs 'he/she' in English)
In the later lesson on animals, though, they are using the animate forms. Either this is a mistake or birds are singled out from other animals.
The examples in the wiki use animate pronouns for animals including hontes, worms, and dragons. So I'm thinking these may be mistakes; at the very least, I think we're justified in flagging them for review.
In the begining of the lesson it is writing that animals depends on the speaker to be consider animate or inanimate.
I had the same question at least twice in this exercise. I feel like animals should be considered animate, and therefore b-class nouns.
Perhapse there could be a nuance of living vs not for animals? If you use k-class the bird is "not moving" or not alive and vise versa for b-class.