Here is what I've learned, which may still be incorrect:
(a) It is not legal to hunt game birds or game animals in Washington with an air rifle.
(b) It is legal to hunt non game animals and non game birds (predatory birds) with an air rifle.
(c) The laws do not make complete sense, because many air rifles are quite capable.
(d) The laws are misunderstand by many, including a significant number of enforcement officers.
"Buyer beware".